Open Collections will be undergoing maintenance Monday June 8th, 2020 11:00 – 13:00 PT. No downtime is expected, but site performance may be temporarily impacted.

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Towards a consensus of opinion Genest, Christian 1983

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1983_A1 G46.pdf [ 6.18MB ]
Metadata
JSON: 831-1.0080161.json
JSON-LD: 831-1.0080161-ld.json
RDF/XML (Pretty): 831-1.0080161-rdf.xml
RDF/JSON: 831-1.0080161-rdf.json
Turtle: 831-1.0080161-turtle.txt
N-Triples: 831-1.0080161-rdf-ntriples.txt
Original Record: 831-1.0080161-source.json
Full Text
831-1.0080161-fulltext.txt
Citation
831-1.0080161.ris

Full Text

TOWARDS A CONSENSUS OF OPINION by CHRISTIAN GENEST B.Sp.Sc, U n i v e r s i t e du Quebec a C h i c o u t i m i , 1977 M.Sc, U n i v e r s i t e de Montreal, 1978 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Mathematics We accept t h i s t h e s i s as conforming to the r e q u i r e d standards THE UNIVERSITY OF BRITISH COLUMBIA January 1983 © C h r i s t i a n Genest, 1983 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the r e q u i r e m e n t s f o r an advanced deg ree a t the U n i v e r s i t y of B r i t i s h C o l u m b i a , I ag ree t h a t t he L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g ree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s t h e s i s f o r s c h o l a r l y pu rpo se s may be g r a n t e d by the Head of my Department or by h i s or he r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department of Ma themat i c s The U n i v e r s i t y of B r i t i s h Co l umb i a 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5 D a t e : J a n u a r y 25, 1983 i i i A b s t r a c t T h i s t h e s i s addresses the problem of combining the p r i o r d e n s i t y f u n c t i o n s , f f , of n i n d i v i d u a l s . In the f i r s t of n two p a r t s , v a r i o u s systems of axioms are developed which c h a r a c t e r i z e s u c c e s s i v e l y the l i n e a r o p i n i o n p o o l , A ( f 1 r . . . , f ) n n = I w f , and the l o g a r i t h m i c o p i n i o n p o o l , G ( f f ) = i=1 i i n n a ( i ) n a ( i ) n f /JUt dy. I t i s f i r s t shown th a t A i s the only i=1 i i=1 i p o o l i n g operator, T ( f f ), which i s e x p r e s s i b l e as n T ( f 1 f . . . , f )(6) = H ( f , ( 6 ) , . . . , f (6),e) f o r some f u n c t i o n H which n n i s continuous i n i t s f i r s t n v a r i a b l e s and s a t i s f i e s H(O,...,O,0) = 0 f o r n- almost a l l 6. The r e g u l a r i t y c o n d i t i o n on H may be dispensed with i f H does not depend on 6. T h i s r e s u l t leads to an i m p o s s i b i l i t y theorem i n v o l v i n g Madansky's axiom of E x t e r n a l B a y e s i a n i t y . Other consequences of t h i s axiom of group r a t i o n a l i t y are a l s o examined i n some d e t a i l and y i e l d a c h a r a c t e r i z a t i o n of G as the only E x t e r n a l l y Bayesian p o o l i n g operator of the form T ( f , , . . . , f ){6) = H ( f , ( 6 ) , . . . , f ( 6 ) ) / n n n JH ( f f )du f o r some H : ( 0 , » ) — > ( 0 , = ° ) . To prove t h i s n r e s u l t , i t i s necessary to i n t r o d u c e a " r i c h n e s s " c o n d i t i o n on the u n d e r l y i n g space of events, ( 0 , M ) . Next, each o p i n i o n f i s i i v regarded as c o n t a i n i n g some " i n f o r m a t i o n " about 0 and we look f o r a p o o l i n g operator whose expected i n f o r m a t i o n content i s a maximum. The operator so ob t a i n e d depends on the d e f i n i t i o n which i s chosen; f o r example, K u l l b a c k - L e i b l e r ' s d e f i n i t i o n e n t a i l s the l i n e a r o p i n i o n p o o l , A. In the second p a r t of the d i s s e r t a t i o n , i t i s argued that the domain of p o o l i n g o p e r a t o r s should extend beyond d e n s i t i e s . The n o t i o n of p r o p e n s i t y f u n c t i o n i s in t r o d u c e d and examples are given which motivate t h i s g e n e r a l i z a t i o n ; these i n c l u d e the well-known problem of combining P-values. A theorem of A c z e l i s adapted to d e r i v e a l a r g e c l a s s of p o o l i n g formulas which encompasses both A and G. A f i n a l c h a r a c t e r i z a t i o n of G i s given v i a the i n t e r p r e t a t i o n of b e t t i n g odds, and the p a r a l l e l between our approach and Nash's s o l u t i o n to the "bargaining problem" i s d i s c u s s e d . James V. Zidek T h e s i s s u p e r v i s o r V Table of Contents A b s t r a c t i i i L i s t of f i g u r e s v i T e c h n i c a l note v i i Acknowledgements v i i i CHAPTER I. PROLEGOMENA 1.1 I n t r o d u c t i o n 1 1.2 The problem of the panel of experts 3 1.3 Previous p r o p o s a l s 6 1.4 O u t l i n e of subsequent chapters 14 CHAPTER I I . POOLING DENSITIES 2.1 Fundamentals and n o t a t i o n 17 2.2 McConway's work i n review 20 2.3 A c h a r a c t e r i z a t i o n of the l i n e a r o p i n i o n pool v i a l o c a l i t y 29 2.4 Seeking E x t e r n a l l y Bayesian procedures 47 2.5 Information maximizing and divergence minimizing p o o l i n g operators 66 2.6 D i s c u s s i o n 81 CHAPTER I I I . POOLING PROPENSITIES 3.1 M o t i v a t i o n 89 3.2 A c l a s s of l o c a l p o o l i n g o p e r a t o r s 99 3.3 D e r i v i n g the l o g a r i t h m i c o p i n i o n pool 124 CHAPTER IV. SUGGESTIONS FOR FURTHER RESEARCH 136 REFERENCES 140 L i s t of Figures Two opinions with a d i f f e r e n t entropy but giving the same pro b a b i l i t y to the true value of the quantity of interest before i t i s revealed to be one v i i T e c h n i c a l Note Th i s t h e s i s was prepared on the Amdahl 470 V8 computer of the U n i v e r s i t y of B r i t i s h Columbia with the a i d of the FMT t e x t -p r o c e s s i n g language. Because the c h a r a c t e r s e t s f o r the Xerox 9700 p r i n t e r are somewhat l i m i t e d , i t was necessary to depart s l i g h t l y from some c o n v e n t i o n a l mathematical symbolism. For in s t a n c e , the l e t t e r "R" had to be res e r v e d to denote the r e a l l i n e and the B r i t i s h Pound S t e r l i n g symbol "£" was s u b s t i t u t e d for s c r i p t e l l . On some oc c a s i o n s , i t was a l s o necessary to wri t e s u b s c r i p t s on the same l i n e as the indexed q u a n t i t i e s , e.g. w(i) i n s t e a d of w when t h i s q u a n t i t y appeared as an i exponent. Furthermore, t i l d a s were s y s t e m a t i c a l l y p r i n t e d over the v a r i a b l e s i n s t e a d of under. We hope the reader w i l l not be inconvenienced by these departures from common usage. The m a t e r i a l i s d i v i d e d i n t o 4 cha p t e r s , and each chapter i n t o s e v e r a l s e c t i o n s . E q uations, d e f i n i t i o n s , theorems and examples are numbered i n the decimal n o t a t i o n . Thus, Equation (2.3.5) r e f e r s to the f i f t h l a b e l l e d equation of S e c t i o n 3, i n Chapter 2. Within S e c t i o n 2.3, i t might be r e f e r r e d to simply as Equation ( 5 ) . v i i i Acknowledgements The author wishes to thank h i s s u p e r v i s o r , P r o f . James V. Zidek, f o r suggesting most of the problems t r e a t e d i n t h i s t h e s i s and f o r f i n a n c i a l a s s i s t a n c e . His a p p r e c i a t i o n i s a l s o extended to Mr. Bruce J . Sharpe f o r h i s constant w i l l i n g n e s s to enter i n t o d i s c u s s i o n . The f i n a n c i a l support of the N a t i o n a l Science and E n g i n e e r i n g Research C o u n c i l and of the U n i v e r s i t y of B r i t i s h Columbia are g r a t e f u l l y acknowledged. ix Dedicat ion T h i s d i s s e r t a t i o n i s d e d i c a t e d with a l l my love to the two women who have supported me throughout my s t u d i e s : my mother, L u c i e Lapointe-Genest, and my wife, C h r i s t i n e Simard-Genest. 1 I. PROLEGOMENA 1 .1 I n t r o d u c t i o n In t h i s t h e s i s , we s h a l l be concerned with the problem of d e v i s i n g methods f o r aggregat i n g o p i n i o n s . By " o p i n i o n , " we mean the ex p r e s s i o n of a person's b e l i e f v i s - a - v i s the outcome of an u n c e r t a i n event, as opposed to an i n t e n t i o n , as i n "op i n i o n p o l l s " (e.g., "whom do you inten d to vote for?".). U s u a l l y , o p i n i o n s w i l l be expressed as ( s u b j e c t i v e ) p r o b a b i l i t y d i s t r i b u t i o n s over the a p p r o p r i a t e space © of " s t a t e s of nature." They might be p r i o r , p o s t e r i o r , s t r u c t u r a l (Fraser 1966) or f i d u c i a l d i s t r i b u t i o n s , f o r i n s t a n c e . Indeed, pro v i d e d that the "degrees of b e l i e f " of an i n d i v i d u a l are assessed q u a n t i t a t i v e l y and i n a coherent manner, they can be shown to conform to the axioms of p r o b a b i l i t y theory (de F i n e t t i 1937; t h i s q u e s t i o n has r e c e n t l y been reexamined by L i n d l e y 1982) . However, common o b s e r v a t i o n and experimental s t u d i e s (Winkler 1967; Tversky & Kahneman 1974; S l o v i c et a l . 1977) tend to c o n f i r m that although an i n d i v i d u a l may have a good knowledge of the r e l a t i v e l i k e l i h o o d of the v a r i o u s p o s s i b l e s t a t e s of nature, i t cannot g e n e r a l l y be expected that he w i l l a l s o master the c a l c u l u s of p r o b a b i l i t i e s and express h i s o p i n i o n 2 a c c o r d i n g l y . Moreover, we- would l i k e to account f o r the use of such widely spread e x p r e s s i o n s of b e l i e f as improper, vague, and uniform or non-informative p r i o r s , as w e l l as the more recent concept of b e l i e f f u n c t i o n d i s c u s s e d by Shafer (1976). T h e r e f o r e , we s h a l l take an o p i n i o n to be any f u n c t i o n f :0 —>[0,=°) on the space 0 of p o s s i b l e s t a t e s of nature. I f the range of f i s r e s t r i c t e d to (0,<»), we w i l l c a l l i t a p r o p e n s i t y f u n c t i o n , or P - f u n c t i o n f o r s h o r t . Furthermore, i f f belongs to a c o l l e c t i o n A = {g:0 —>[0,»)|fgdM =1} of p r o b a b i l i t y d e n s i t i e s with r e s p e c t to a dominating measure v on 0, we w i l l say that f i s a ^ - d e n s i t y . In accordance with the l i t e r a t u r e , we w i l l c a l l a s s e s s o r or expert any person who i s asked to produce h i s o p i n i o n concerning 0. G e n e r a l l y , an a s s e s s o r w i l l be some kind of expert whose o p i n i o n on the subject matter i s deemed to be e n l i g h t e n i n g , but i t need not be so. In our d i s c u s s i o n , i t w i l l always be assumed t h a t when asked f o r t h e i r o p i n i o n , the e x p e r t s are capable and w i l l i n g to present an assessment of a l l r e l e v a n t f a c t s and evidence known to them. As long as t h i s i s the case, every o p i n i o n has i t s value and should be t r e a t e d with c o n s i d e r a t i o n . For, in s u b j e c t i v e or p e r s o n a l i s t i c p r o b a b i l i t y theory, the r e l a t i v e l i k e l i h o o d a t t r i b u t e d to an event or hypothesis i s simply what the assessor b e l i e v e s i t to be; there i s no such t h i n g as a " c o r r e c t " or " o b j e c t i v e " o p i n i o n . For a c r i t i c a l review of t h i s approach, c f . Fine (1973). 3 1.2 The problem of the panel of experts In a d e c i s i o n a n a l y s i s , i t i s o f t e n necessary to combine a group of i n d i v i d u a l s ' b e l i e f s i n t o a s i n g l e r e p r e s e n t a t i v e o p i n i o n which may be thought of as a consensus of these peoples' judgements. T h i s problem of determining d e c i s i o n r u l e s f o r s t a t i s t i c a l l y a ggregating i n d i v i d u a l o p i n i o n s without group d i s c u s s i o n nor b a r g a i n i n g i s c a l l e d the problem of the panel of  e x p e r t s , a f t e r R a i f f a (1968). Let us suppose, f o r i n s t a n c e , that a d e c i s i o n maker who has very l i t t l e knowledge of a subject-matter i s c o n f r o n t e d with the need to q u a n t i f y h i s b e l i e f s and determine a p r i o r d i s t r i b u t i o n before he can undertake a formal Bayesian a n a l y s i s . To s o l v e t h i s problem, he might choose to express h i s ignorance by using a p r i o r d i s t r i b u t i o n which adds l i t t l e to the sample i n f o r m a t i o n ; much work has been done along these l i n e s by L i n d l e y (1961), J e f f r e y s (1967), Novick & H a l l (1965) and Z e l l n e r (1977). However, the u n c r i t i c a l use of non-informative p r i o r measures sometimes leads to i n c o n s i s t e n c y and paradoxes such as those presented by Dawid, Stone & Zidek (1973) or Stone (1976). Moreover, t h i s approach w i l l not be s a t i s f a c t o r y i f the problem at hand i s of major importance and i t s a n a l y s i s w i l l r e s u l t i n a c o s t l y , i r r e v e r s i b l e d e c i s i o n . I f time i s p r e s s i n g or c o l l e c t i o n of a l a r g e amount of data i s i m p r a c t i c a l , the d e c i s i o n maker may w e l l decide to c o n s u l t one or s e v e r a l people who do have knowledge b e l i e v e d to be r e l e v a n t to the q u e s t i o n , 4 i . e . e x p e r t s . These e x p e r t s are s a i d to form a panel of c o n s u l t a n t s . Even a f t e r e x t e n s i v e d i s c u s s i o n amongst them about t h e i r b e l i e f s and proper m o d i f i c a t i o n of t h e i r r e s p e c t i v e o p i n i o n s to take i n t o account a l l the j o i n t l y p e r c e i v e d and a v a i l a b l e i n f o r m a t i o n , i t i s u n l i k e l y that the experts w i l l converge to a s t a t e of t o t a l agreement. When t h i s happens, we say that the group i s l e f t i n d i s s e n s u s . As only one o p i n i o n i s needed in the end, how does the d e c i s i o n maker proceed to e x t r a c t i t from the number of ( p o s s i b l y ) d i v e r g i n g o p i n i o n s he has c o l l e c t e d , without being i r r e s p o n s i v e to any p a r t i c u l a r assessment? Des p i t e the prevalence of c o n s u l t i n g i n c o u n t l e s s d e c i s i o n -making s i t u a t i o n s , t h i s problem has r e c e i v e d comparatively l i t t l e a t t e n t i o n i n the l i t e r a t u r e , as p o i n t e d out i n the review papers of Winkler (1968) and Hogarth (1975,1977). An i n s t r u c t i v e i n t r o d u c t i o n to the v a r i o u s t h e o r e t i c a l q u e s t i o n s r a i s e d by group-assessments i s p r o v i d e d by R a i f f a (1968). In c o n c e p t u a l i z i n g the phenomenon, we w i l l assume that a l l d i s c u s s i o n and argument have taken p l a c e at the time when the d e c i s i o n maker i s presented with a number, n, of o p i n i o n s , one f o r each member of the panel of e x p e r t s . We w i l l p l a c e o u r s e l v e s i n h i s shoes and attempt to f i n d d e s i r a b l e p r o p e r t i e s which a consensus o p i n i o n should have. More s p e c i f i c a l l y , we i n t e n d to propose and e x p l o r e the consequences of v a r i o u s i n t e r p r e t a t i o n s of such vague concepts as "adequacy," 5 " r e p r e s e n t a t i v e n e s s " and "consensus." The approach we take here i s normative, i n the sense that we p r e s c r i b e the way i n which a d e c i s i o n maker should process expert o p i n i o n s i f he wishes to adhere to c e r t a i n p o s t u l a t e s of coherence and r a t i o n a l i t y . No attempt i s made to d e s c r i b e how a person c o n f r o n t e d with the a c t u a l problem would be observed to c a r r y out the t a s k . Moreover, we are i m p l i c i t l y adopting the view that i n the presence of u n c e r t a i n t y about ©, the q u a n t i t y of i n t e r e s t to the d e c i s i o n maker i s T ( f f ), the consensus n o p i n i o n d e s c r i b i n g h i s f i n a l assessment of b e l i e f s concerning the p o s s i b l e outcomes of the event upon c o l l e c t i n g the experts* views f f . n T h i s a t t i t u d e i s by no means g e n e r a l l y accepted, although i t has some su p p o r t e r s (Winkler 1968; Weerahandi & Zidek 1978; Bernardo 1979 i n a d i f f e r e n t c o n t e x t ) . However, i t makes p e r f e c t l y c l e a r t h at the problem we propose to examine d i f f e r s markedly from that of a group faced with a decision-making s i t u a t i o n , where the main concern l i e s with the f i n a l group d e c i s i o n and where -by way of n e c e s s i t y - c o n s i d e r a t i o n s of u t i l i t y and b a r g a i n i n g are invoked. The d i s t i n c t i o n s and r e l a t i o n s between these two problems have been ... w.ell emphasized in a survey paper by Weerahandi & Zidek (1981). 6 Admittedly, t h e r e f o r e , the e x p r e s s i o n " d e c i s i o n maker" i s something of a misnomer. In f a c t , the present set-up leaves open the p o s s i b i l i t y that the d e c i s i o n maker rep r e s e n t s the " s y n t h e t i c p e r s o n a l i t y " of a group amongst whose members d i s c u s s i o n has f a i l e d to c r e a t e a consensus, but which i s c a l l e d upon, n e v e r t h e l e s s , to produce a s i n g l e assessment of b e l i e f s r e p r e s e n t a t i v e of the v a r i o u s o p i n i o n s v o i c e d . T h i s would be the case i f , f o r example, a group of m e t e o r o l o g i s t s was asked to issue a j o i n t f o r e c a s t . I t i s important to r e a l i z e , then, that the r o l e played by our d e c i s i o n maker has a l o t i n common with that of the s t a t i s t i c i a n of standard d e c i s i o n theory when he attempts to devise s t r a t e g i e s to estimate a q u a n t i t y from a number of o b s e r v a t i o n s . Only here, o b s e r v a t i o n s are o p i n i o n s , the estimate sought i s a l s o an o p i n i o n , and there i s no "true o b j e c t i v e o p i n i o n " a g a i n s t which to judge the performance of the p o o l i n g formula that the d e c i s i o n maker chooses to use. 1.3 P r e v i o u s p r o p o s a l s The l i t e r a t u r e which r e l a t e s to the s o - c a l l e d problem of the panel of experts d i v i d e s roughly i n t o two main streams: on the one hand are those papers which d e a l d i r e c t l y with p e r s o n a l p r o b a b i l i t y d i s t r i b u t i o n s and t h e i r a g g r e g a t i o n ; on the other are those which are concerned with a broader p i c t u r e , that of a 7 group d e c i s i o n problem, and which focus on c o n s i d e r a t i o n s of u t i l i t y r a t h e r than on the consensus of o p i n i o n s . References of the l a t t e r type w i l l only be mentioned when they i n c l u d e some h e l p f u l comments r e g a r d i n g the problem at hand. F o l l o w i n g Stone (1961), most i n v e s t i g a t o r s have re p r e s e n t e d group assessments by t a k i n g a weighted average of i n d i v i d u a l d i s t r i b u t i o n s . F o r m a l l y , i f f r e p r e s e n t s the p r o b a b i l i t y i d e n s i t y of the i - t h member of the group concerning the unknown q u a n t i t y 6, then the l i n e a r o p i n i o n pool f o r n e x p e r t s i s d e f i n e d by n T ( f f ) = Z w f , (1.3.1) n i = 1 i i n where w > 0 and I w = 1 . The r e s t r i c t i o n s on the weights, i i = 1 i w , i n s u r e that the j o i n t o p i n i o n of the group, T ( f f ), i n w i l l be a d e n s i t y . Winkler (1968) d i s c u s s e s the problem of determining the weights f o r each expert's d i s t r i b u t i o n and proposes v a r i o u s ad hoc s o l u t i o n schemes. In subsequent s t u d i e s , he and o t h e r s ( S t a e l Von H o l s t e i n 1970; Winkler 1971) found that the uniform weighting scheme, w = 1/n, was never outperformed more than i m a r g i n a l l y ( i n terms of p r e d i c t i v e a b i l i t y ) by other schemes 8 which attempted to rank each expert a c c o r d i n g to h i s e x p e r t i s e or past performance (these i n c l u d e d the method of Roberts (1965) f o r combining and updating weights u s i n g Bayes' Theorem). A l s o s u p p o r t i n g Formula (1.3.1) i s the o f t e n observed e m p i r i c a l f a c t that composite d i s t r i b u t i o n s show g r e a t e r p r e d i c t i v e a b i l i t y than most of the i n d i v i d u a l e x p e r t s , 1 a phenomenon which might be l i n k e d to the r e l i a b i l i t y of average p o i n t estimates i n the c l a s s i c a l theory of e s t i m a t i o n . But i t was i n the work of Bacharach (1973, 1975) that s o l i d t h e o r e t i c a l grounds f o r implementing the l i n e a r o p i n i o n pool began to emerge. Although Bacharach i s mainly concerned with the e x i s t e n c e of a s e n s i b l e group p r e f e r e n c e r e l a t i o n f o r o r d e r i n g the p o s s i b l e courses of a c t i o n i n the face of d i f f e r e n c e s of o p i n i o n and u t i l i t y , he f i n d s c o n d i t i o n s on t h i s group p r e f e r e n c e r e l a t i o n which e n t a i l , as a unique s o l u t i o n , Stone's Formula (1.3.1). In a theorem which he a t t r i b u t e s to Madansky (1964), Bacharach d e r i v e s (1.3.1) by assuming e s s e n t i a l l y that the group w i l l p r e f e r an a c t i o n A to another a c t i o n B whenever each of i t s members does, and that the group's p r e f e r e n c e r e l a t i o n i s not a f f e c t e d by the presence of i r r e l e v a n t a l t e r n a t i v e s . A f t e r a r g u i n g in favour of these 1 Dr. F. P. G l i c k brought to my a t t e n t i o n some recent work of Alan Shapiro (1977,1979) who r e d i s c o v e r e d t h i s f a c t f o r himsel f and used the l i n e a r o p i n i o n pool to i n c r e a s e d i a g n o s t i c accuracy of p h y s i c i a n - e x p e r t s . 9 p o s t u l a t e s , he goes on to show that i f an e x t r a c o n d i t i o n which he c a l l s group r a t i o n a l i t y i s introduced, the pool can be forced i n t o d i c t a t o r i a l form, i . e . one of the w 's of (1.3.1) equals 1 i while the remainder are 0. This f a r - r e a c h i n g c o n d i t i o n a s s e r t s that the group acts as i f i t were a s i n g l e e x p e c t e d - u t i l i t y maximizer. In Section 2.4, we too w i l l prove such an " i m p o s s i b i l i t y theorem" using a s i m i l a r concept c a l l e d by Madansky (1964,1978) E x t e r n a l B a y e s i a n i t y . In our endeavour, we have been very much stimulated by the work of Kevin McConway (1981) who was the f i r s t to give a strong j u s t i f i c a t i o n for using the l i n e a r o p i n i o n pool w i t h i n the framework we propose to adopt ourselves. His main theorem s t a t e s that i f a d e c i s i o n maker wants h i s process of consensus f i n d i n g to commute with the m a r g i n a l i z a t i o n of the d i s t r i b u t i o n s i n v o l v e d , then he has no a l t e r n a t i v e but to use Formula (1.3.1). Chapter 2 of t h i s t h e s i s w i l l s t a r t with a d i s c u s s i o n of McConway's r e s u l t . Despite i t s great p o p u l a r i t y , the weighted average formula i s endowed with features which may i n c e r t a i n circumstances be viewed as drawbacks. For i n s t a n c e , Winkler (1968) notes that (1.3.1) i s t y p i c a l l y multi-modal on i t s domain and so may f a i l to i d e n t i f y a parameter which t y p i f i e s the i n d i v i d u a l choices. This leads him to formulate an a l t e r n a t e p r e s c r i p t i o n , which he c a l l s the n a t u r a l conjugate approach. In t h i s method, each group member's opinion i s deemed to c o n s t i t u t e "sample evidence" 10 which can be represented by a n a t u r a l conjugate p r i o r ( B i c k e l & Doksum 1977, p. 77) to the d i s t r i b u t i o n of the d a t a - g e n e r a t i n g process of i n t e r e s t . In order to form the group assessment, t h e r e f o r e , the d e c i s i o n maker need on l y combine o p i n i o n s i n a manner s i m i l a r to s u c c e s s i v e a p p l i c a t i o n s of Bayes' Theorem. The reader w i l l r ecognize that even i f t h i s approach o b v i a t e s the p o s s i b i l i t y of multi-modal d i s t r i b u t i o n s , i t leaves us nonetheless with the d i f f i c u l t q u e s t i o n of determining weights fo r each of the e x p e r t s , as w e l l as the degree to which t h e i r o p i n i o n s are based on o v e r l a p p i n g experience and data sources. I t i s p r e c i s e l y these d i f f i c u l t i e s which brought M o r r i s (1974,1977) to e l a b o r a t e a theory of expert use that i s e n t i r e l y c o n s i s t e n t with the Bayesian p h i l o s o p h y . In h i s work, M o r r i s pushes Winkler's idea one step f u r t h e r and t r e a t s each expert p r o b a b i l i t y d i s t r i b u t i o n as a random v a r i a b l e whose value i s r e v e a l e d to the d e c i s i o n maker. To o b t a i n the consensus d i s t r i b u t i o n , the d e c i s i o n maker must then proceed to i n t r o s p e c t a l i k e l i h o o d f u n c t i o n r e p r e s e n t i n g h i s assessment of the d i f f e r e n t e x p e r t s ' knowledge and combine t h e i r o p i n i o n s one-by-one with h i s own u s i n g Bayes' r u l e . A c o n v e n t i o n a l , a l b e i t complex uni-Bayesian a n a l y s i s r e s u l t s . In the simplest case, that where the members of the panel r e l y on independent data bases and are good p r o b a b i l i t y a s s e s s o r s ( i . e . they are 11 c a l i b r a t e d 1 ), M o r r i s shows that the normalized product of the e x p e r t s ' p r i o r s o b t a i n s : n a( i ) n a( i ) T ( f 1 f . . . , f ) = n f / ; n f d*x, ( 1 . 3 . 2 ) n i=1 i i= 1 i with a ( i ) = 1,1£i£n; t h i s i s a p a r t i c u l a r case of the s o - c a l l e d l o g a r i t h m i c o p i n i o n pool which Dalkey (1975) had e a r l i e r proposed on an ad hoc b a s i s . Note that f o r T ( f f ) to be a n M - d e n s i t y i n ( 1 . 3 . 2 ) , the a ( i ) ' s do not need to add up to one. However, M o r r i s assumes that the d e c i s i o n maker i s one of the members of the panel, which f o r c e s a ( l ) = ... = a ( n ) ; otherwise, the s o l u t i o n would be expert-dependent and thus c o u l d not c o n s t i t u t e an a c c e p t a b l e consensus. Although M o r r i s ' work c e r t a i n l y i s c o n c e p t u a l l y a p p e a l i n g , c o n s i s t e n t and i n s i g h t f u l , only few would agree with him that i t r e p r e s e n t s a p r a c t i c a l methodology because of the insurmountable assessment problems which o v e r l a p p i n g experience would cause. For t h i s reason, other r e s e a r c h e r s have continued to seek d i r e c t ways of p o o l i n g o p i n i o n s . It i s i n t e r e s t i n g to observe that the p r e s c r i p t i o n embodied 1 As L i n d l e y & a l . (1979) observed, t h i s i s a s k i n g a l o t . On the other hand, Dawid (1982) has r e c e n t l y shown that a coherent Bayesian expects to be w e l l c a l i b r a t e d ! ! 1 2 in (1.3.2) i s not only a n a t u r a l i m p l i c a t i o n of M o r r i s ' a n a l y s i s but as w e l l i t i s E x t e r n a l l y Bayesian when the a ( i ) ' s add up to one. T h i s axiom, formulated by Madansky (1964,1978), r e q u i r e s the commutativity of the o p e r a t i o n s of ( i ) compounding i n d i v i d u a l p r o b a b i l i t i e s i n t o a group p r o b a b i l i t y ; and ( i i ) updating p r o b a b i l i t y assessments v i a Bayes' formula. The ba s i c p r o p e r t i e s of E x t e r n a l l y Bayesian procedures have been e x p l o r e d by Madansky (1978) who finds, that of those p o o l i n g formulas which are d i s c u s s e d i n the l i t e r a t u r e , only d i c t a t o r s h i p s , (1.3.2) and c e r t a i n a p p l i c a t i o n s of the n a t u r a l conjugate approach s u r v i v e the t e s t . E x t e r n a l B a y e s i a n i t y has a l s o been supported by Weerahandi & Zidek (1978), who c a l l i t p r i o r - t o - p o s t e r i o r coherence. T h e i r unpublished manuscript c o n t a i n s an i n c o r r e c t (but c o r r e c t a b l e , c f . Theorem 2.4.6) d e r i v a t i o n of the l o g a r i t h m i c o p i n i o n pool using E x t e r n a l B a y e s i a n i t y . Moreover, they argue that randomized d e c i s i o n r u l e s need to be int r o d u c e d because, i n some cases, the p r i o r o p i n i o n s of the experts are so d i s c r e p a n t that " t o s s i n g a c o i n " i s the only s a t i s f a c t o r y means f o r choosing between them. In S e c t i o n 2.5, a s i m i l a r idea w i l l l e a d us to s t i l l another c h a r a c t e r i z a t i o n of Stone's l i n e a r o p i n i o n p o o l , w h i l s t i t w i l l be shown i n S e c t i o n 2.4 that the l o g a r i t h m i c o p i n i o n pool i s , i n some sense, the only E x t e r n a l l y Bayesian procedure f o r combining the p r i o r d e n s i t y f u n c t i o n s f f of n n i n d i v i d u a l s . 13 Our survey of the l i t e r a t u r e on the problem of p o o l i n g d e n s i t i e s would not be complete without at l e a s t a b r i e f mention of the two f o l l o w i n g papers. In 1959, E i s e n b e r g & Gale (see a l s o N o r v i g 1967) presented an ingenious scheme f o r combining p r o b a b i l i t y d i s t r i b u t i o n s based on the "pari-mutuel" b e t t i n g method. T h e i r idea was that the p r i n c i p l e s determining " t o t a l i s a t o r odds" c o u l d a p p r o p r i a t e l y determine group judgements i n more general s i t u a t i o n s . However, i t can be shown that c e r t a i n i n d i v i d u a l o p i n i o n s w i l l allow t h e i r h o l d e r s to d i c t a t e the consensus odds, and f o r that reason, the pari-mutuel method has never been very p o p u l a r . Then, DeGroot (1974) proposes that upon being a p p r i s e d of the d i s t r i b u t i o n s of the other members of the panel, each expert updates h i s own p r i o r u s ing Formula (1.3.1) by a s s i g n i n g importance weights to himself and h i s peers. The procedure i s then i t e r a t e d u n t i l f u r t h e r r e v i s i o n s no longer a l t e r any of the members' o p i n i o n s , and l i m i t theorems from Markov Chain theory are invoked to determine when a consensus d i s t r i b u t i o n e x i s t s and what i t i s when i t does e x i s t . Berger (1981) p o i n t s out an e r r o r i n DeGroot's o r i g i n a l paper and g i v e s the exact c o n d i t i o n s under which the i t e r a t i v e process w i l l converge. A v a r i a t i o n on the theme i s d e s c r i b e d by Press (1978), who a l s o p r o v i d e s an e x t e n s i v e l i s t of r e f e r e n c e s . 14 Although DeGroot's method does not f o r m a l l y f i t our set-up (remember that we assume a l l d i s c u s s i o n i s c l o s e d ) , i t c o u l d be imagined that the i t e r a t i v e process i s c a r r i e d out by the d e c i s i o n maker h i m s e l f , once he has taken note of the v a r i o u s o p i n i o n s expressed as w e l l as the r a t i n g s that each i n d i v i d u a l expert granted to h i m s e l f and to the other people who were c o n s u l t e d . Along with French (1981), however, we f i n d three important flaws i n t h i s g e n e r a l approach. They ar e : ( i ) the l i n e a r o p i n i o n pool i s s t i l l proposed as an ad hoc procedure; ( i i ) i t i s assumed that no o u t s i d e data, o b s e r v a t i o n s or i n f o r m a t i o n about the value of 8 i s a v a i l a b l e ( t h i s i n v a l i d a t e s E x t e r n a l B a y e s i a n i t y as a s e l e c t i o n p r i n c i p l e ) ; and l a s t but not l e a s t , ( i i i ) no p r o v i s i o n i s made f o r the ( n o n - n e g l i g i b l e ) case when the i t e r a t i v e procedure leaves the group in d i s s e n s u s . 1.4 O u t l i n e of subsequent chapters As p o i n t e d out i n the f i r s t s e c t i o n of t h i s chapter, we do not r e s t r i c t o u r s e l v e s to what c o u l d be c a l l e d the c l a s s i c a l form of the problem of the panel of e x p e r t s , where o p i n i o n s are assumed to have been expressed as p r o b a b i l i t y d i s t r i b u t i o n s over 0. Rather, we enlarge the d e f i n i t i o n of o p i n i o n to i n c l u d e any e x p r e s s i o n of b e l i e f s , f:0 — > [ 0 , » ) , i n t e g r a b l e or not. Thus, we w i l l be concerned with p r o p e n s i t y f u n c t i o n s as w e l l as d e n s i t y f u n c t i o n s . The two t o p i c s w i l l be d i s c u s s e d i n separate c h a p t e r s . 15 S t a r t i n g with the problem of p o o l i n g d e n s i t i e s , Chapter 2 begins with a d i s c u s s i o n of McConway's (1981) c h a r a c t e r i z a t i o n of the l i n e a r o p i n i o n pool v i a the M a r g i n a l i z a t i o n P r o p e r t y . A short proof of h i s theorem appears in S e c t i o n 2.2. We are then l e d to propose i n S e c t i o n 2.3 a d i f f e r e n t d e r i v a t i o n of the same operator founded on the concept of " l o c a l i t y . " At l e a s t one avenue f o r g e n e r a l i z a t i o n i s e x p l o r e d . S e c t i o n 2.4 i s devoted to a study of Madansky's (1964;1978) axiom of E x t e r n a l B a y e s i a n i t y and the l o g a r i t h m i c o p i n i o n pool i s shown to be the only q u a s i - l o c a l p o o l i n g formula which i s c o n s i s t e n t with t h i s p o s t u l a t e . Some of the g e n e r a l p r o p e r t i e s possessed by E x t e r n a l l y Bayesian procedures are d e r i v e d on the way. In S e c t i o n 2.5, we regard a consensus d i s t r i b u t i o n as a s t a t i s t i c which condenses the i n f o r m a t i o n contained i n a set of o p i n i o n s . The idea of an i n f o r m a t i o n maximizing p o o l i n g operator then leads to yet another c h a r a c t e r i z a t i o n of Stone's l i n e a r o p i n i o n p o o l . Furthermore, a l a r g e c l a s s of p o o l i n g formulas c o n t a i n i n g both the l i n e a r and the l o g a r i t h m i c pools as l i m i t i n g cases i s d e r i v e d u s i n g K u l l b a c k ' s (1968) n o t i o n of divergence. F i n a l l y , S e c t i o n 2.6 d i s c u s s e s these f i n d i n g s and r e i t e r a t e s some words of c a u t i o n . Chapter 3 addresses the more gen e r a l problem of combining a number of p r o p e n s i t y f u n c t i o n s . Examples are presented i n S e c t i o n 3.1 which motivate t h i s g e n e r a l i z a t i o n ; amongst them appears the well-known problem of combining independent t e s t s of 1 6 h y p o t h e s i s . A l l these examples are continued i n S e c t i o n 3.2, where we focus our a t t e n t i o n on p o o l i n g o p e r a t o r s which are l o c a l . Here, we argue that the q u a s i - a r i t h m e t i c weighted means of Hardy, L i t t l e w o o d & P6lya (1934) are the only " s e n s i b l e " l o c a l r u l e s f o r aggregating p r o p e n s i t i e s ; the approach i s axiomatic and w e l l s u i t e d to those cases where the expe r t s ' s c a l e s of b e l i e f are intercomparable. The l a s t s e c t i o n i s devoted to c h a r a c t e r i z i n g the l o g a r i t h m i c pool when the c o m p a r a b i l i t y assumption i s not met; the p a r a l l e l between our approach and Nash's (1950) s o l u t i o n to the b a r g a i n i n g problem i s a l s o d i s c u s s e d . F i n a l l y , Chapter 4 c o n t a i n s suggestions f o r f u r t h e r r e s e a r c h . 1 7 I I . POOLING DENSITIES 2.1 Fundamentals and n o t a t i o n Throughout t h i s chapter, we w i l l denote by 0 the space of (mutually e x c l u s i v e ) s t a t e s of nature over which each of a group of n a s s e s s o r s (n>2) w i l l be asked to produce a p r o b a b i l i t y d i s t r i b u t i o n . For convenience, we w i l l assume that \i i s a dominating measure on © and that each o p i n i o n i s expressed as a M-measurable f u n c t i o n f:0 —>[0,°°) with f f d n = 1. Because of the p r o p e r t i e s of the Caratheodory process f o r g e n e r a t i n g measures, there i s no l o s s of g e n e r a l i t y i n assuming n to be complete and i n t a k i n g Q ( M ) = { A c © | M ( B ) = M ( B n A ) +M ( B \ A ) f o r a l l B c ©} to be the a - f i e l d of M _measurable s e t s . In a p p l i c a t i o n s , u w i l l u s u a l l y be o - f i n i t e , so that every other measure v which i s a b s o l u t e l y continuous with r e s p e c t to ii w i l l have a Radon-Nikodym d e r i v a t ive (Sion 1968, p. 110). But o~ f i n i t e n e s s i s not e s s e n t i a l otherwise. We w i l l w r i t e A f o r the c o l l e c t i o n of a l l ^-measurable d e n s i t i e s on 0, and ( f , , . . . , f ) to represent e i t h e r a t y p i c a l n n n element of A or e l s e the f u n c t i o n f:0 —>[0,°°) d e f i n e d by f(0 ) = (f , ( 0 ) , . . . , f ( 0 ) ) . The i n t e r p r e t a t i o n w i l l always be n c l e a r from the context, so no c o n f u s i o n w i l l a r i s e from t h i s c o n v e n t i o n . I t w i l l a l s o be assumed that A * 0, so that there 18 e x i s t s A e J 2 ( M ) with 0 < y ( A ) < » . n By a p o o l i n g operator on 0, we mean any a p p l i c a t i o n T:A — > A which maps the n-tuple (f f ) to T ( f f ), a u~ n n d e n s i t y . The f o l l o w i n g d e f i n i t i o n l i s t s the p r o p e r t i e s of p o o l i n g o p e r a t o r s which we w i l l most o f t e n r e f e r t o . D e f i n i t i o n 2.1.1 n We say that a p o o l i n g operator T:A —>A i s (1) l o c a l n i f f t h ere e x i s t s a Lebesgue-measurable f u n c t i o n G : [ 0 , = ° ) — > [ 0 , ® ) such that T ( f 1 f . . . , f ) = G n ( f , , . . . , f ) y-a.e. Here, n n n r e p r e s e n t s the usual composition of f u n c t i o n s . (2) q u a s i - l o c a l n i f f there e x i s t s a f u n c t i o n C:A — > ( 0 , = ° ) such that T t f , , . . . ^ )«C(f,,...,f ) i s l o c a l , n n ( 3 ) unanimity p r e s e r v i n g i f f T ( f 1 f . . . , f ) = f y-a.e. whenever f =f y-a.e. f o r a l l n i 1<i<n. (4) dogma p r e s e r v i n g n i f f S u p p ( T ( f f )) c u Supp(f ) y-a.e., where i n n i = 1 i gen e r a l Supp(f) = ( 0 e 0 | f ( 0 ) * 0 } . 19 (5) a d i c t a t o r s h i p i f f there e x i s t s 1^i^n such that T ( f , , . . . , f ) = f M-a.e. f o r a l l c h o i c e s of f , , . . . , f i n n i n the domain of T. To prove theorems, we w i l l o f t e n make use of the f o l l o w i n g elementary r e s u l t s from the Theory of f u n c t i o n a l e q u a t i o n s . T h e i r p r o o f s are to be found i n A c z e l (1966). Lemma 2.1.2 n Let h:R —>R be Lebesgue-measurable or non-decreasing i n each of i t s n v a r i a b l e s . I f h(x)+h(y) = h(x+y) (2.1.1) for a l l x and y, v e c t o r s of r e a l numbers, then there e x i s t n c o n s t a n t s c c e R such that h(x) = l e x over i t s n i=1 i i domain, x = ( x 1 f . . . , x ). The r e s u l t a l s o holds t r u e when the n n domain of h i s [0,°°) or [0,K] with K > 0 a co n s t a n t . Lemma 2.1.3 n Let h:(0,») —>R be Lebesgue-measurable or non-decreasing i n each of i t s n v a r i a b l e s . If h(x)-h(y) = h(x-y) (2.1.2) n c ( i ) for a l l x,y > o, then h(x) = II x f o r some c ( i ) e R. i = 1 i 20 Equations (2.1.1) and (2.1.2) are u s u a l l y r e f e r r e d to as Cauchy's f u n c t i o n a l equation. In the present context, a d d i t i o n and m u l t i p l i c a t i o n of v e c t o r s i s taken to be componentwise. 2.2 McConway's work i n review In t h i s s e c t i o n , McConway's (1981) d e r i v a t i o n of the l i n e a r o p i n i o n pool w i l l be d i s c u s s e d . A short proof of h i s theorem ( l a b e l l e d 2.2.4) w i l l a l s o be o f f e r e d . The d i s c u s s i o n w i l l motivate and serve as a background f o r our own r e s u l t s . It has a l r e a d y been mentioned that the approach adopted by McConway f i t s the d e s c r i p t i o n of the problem of the panel of experts set out i n S e c t i o n 2 of Chapter 1. To j u s t i f y the p r e s c r i p t i o n embodied i n (1.3.1), McConway f i r s t i n t r o d u c e s what c o u l d be c a l l e d the M a r g i n a l i z a t i o n P o s t u l a t e (MP). T h i s c o n d i t i o n s t i p u l a t e s that the same consensus d i s t r i b u t i o n should be a r r i v e d at whether ( i ) the a s s e s s o r s ' d i s t r i b u t i o n s are f i r s t combined and then some m a r g i n a l i z a t i o n i s performed on the consensus; or ( i i ) each a s s e s s o r i n d i v i d u a l l y performs the m a r g i n a l i z a t i o n and the r e s u l t i n g marginal d i s t r i b u t i o n s are pooled i n t o a consensus d i s t r i b u t i o n . If a m i l d c o n d i t i o n tantamount to that i n our d e f i n i t i o n of a dogma p r e s e r v i n g p o o l i n g operator i s added, then McConway 21 proves that the M a r g i n a l i z a t i o n P o s t u l a t e i s e q u i v a l e n t to what he c a l l s the Strong Setwise F u n c t i o n Property (SSFP), which i n turn holds true i f and only i f the p o o l i n g operator i s of the form (1.3.1). (To c a r r y out h i s program, McConway i n t r o d u c e s the "Weak Setwise F u n c t i o n P r o p e r t y " (WSFP) and proceeds to show that t h i s p r o p e r t y i s e q u i v a l e n t to h i s M a r g i n a l i z a t i o n P o s t u l a t e . T h i s f a c t c o n s t i t u t e s Theorem 3.1 of h i s paper. However, the proof i s obscured by a m i s p r i n t . The t h i r d l i n e of the l a s t paragraph, p. 411, should read "any S e l which c o n t a i n s A has oik) as a sub-o-algebra..." and not "any S e l c o n t a i n s A, has o(A) as a sub - a - a l g e b r a . . . " ) . In h i s treatment, McConway does not assume the e x i s t e n c e of a dominating measure y, and consequently h i s r e s u l t s are s t a t e d in terms of p r o b a b i l i t y measures as opposed to d e n s i t i e s . However, McConway does say that " i n p r a c t i c e , the experts w i l l u s u a l l y agree on some obvious a - f i e l d over 0," and we c l a i m that most of the time, a n a t u r a l y w i l l impose i t s e l f j u s t as w e l l , so that l i t t l e i s l o s t by assuming i t s e x i s t e n c e . T h i s p o i n t of view i s not e n t i r e l y new, as an i n s p e c t i o n of the set-up i n Stone (1961) or Weerahandi & Zidek (1978) w i l l c o n f i r m . (Note that f o r y to e x i s t , i t s u f f i c e s f o r the experts to agree on some n a t u r a l a - a d d i t i v e f u n c t i o n T on a r i n g H. The Caratheodory process w i l l then a u t o m a t i c a l l y extend T to y and H w i l l be co n t a i n e d i n fi(y). Cf. Sion 1968, p. 67). 22 When a n a t u r a l dominating measure u on 0 e x i s t s , McConway's SSFP c o n d i t i o n can be, formulated as f o l l o w s : D e f i n i t i o n 2.2.1 (McConway 1981) n A p o o l i n g operator T:A — > A has the Strong Setwise F u n c t i o n n Property (SSFP) i f f there e x i s t s a f u n c t i o n F:[0,1] — > [ 0 , 1 ] such that / I ( A ) - T ( f , , . . . ,f )du = F [ f I ( A ) f ,d/ti, f I ( A ) f d / i ] (2.2.1) n n f o r a l l A e 0 ( M ) , where i n general 1 ( A ) stands f o r the i n d i c a t o r f u n c t i o n of the set A. Before we s t a t e McConway's Theorem, we make a d e f i n i t i o n of our own: 1 D e f i n i t i o n 2.2.2 We say that a space ( 0 ,M ) i s t a n g i b l e i f f there e x i s t (at l e a s t ) three y-measurable neighbourhoods A 1 f A 2 , A 3 i n 0 such that ( I ) 0 < M ( A ) < i = i ,2,3; i 1 S h o r t l y a f t e r the completion of t h i s t h e s i s , a paper of C a r l Wagner (1982) was brought to our a t t e n t i o n i n which the author uses the term " t e r t i a r y space" to denote what we c a l l a t a n g i b l e space. Theorem 7 of h i s paper i s e q u i v a l e n t to the f o r m u l a t i o n of McConway's Theorem presented in Theorem 2.2.4 below. 23 and ( I I ) uik n A ) > 0 => i = j . i j One might wonder what an " i n t a n g i b l e " space looks l i k e . A more or l e s s complete answer to t h i s q u e s t i o n i s provided by the f o l l o w i n g lemma. Lemma 2.2.3 The spaces 0, = {0,}, and 0 2 = {9, ,t52) with v = counting measure t y p i f y the c l a s s of i n t a n g i b l e spaces. P r o o f : Let (®,v) be i n t a n g i b l e , so that t h e r e are at most two y-measurable s e t s A,,A 2 with p r o p e r t i e s (I) and ( I I ) . Assuming A * 0 i s enough to guarantee the e x i s t e n c e of at l e a s t one such set A,. If there e x i s t s only one set A, s a t i s f y i n g (I) & ( I I ) , then the f u n c t i o n f = I ( A 1 ) / ^ i ( A 1 ) i s the only element of A and (Q,a) i s c l e a r l y e q u i v a l e n t to (©,,»>). On the other hand, i f there are e x a c t l y two s e t s A,,A 2 with p r o p e r t i e s (I) and ( I I ) , then i t f o l l o w s from D e f i n i t i o n 2.2.2 that any y-measurable subset B of A must obey y(B) = 0 or n(B) i = n(k ), i = 1 ,2,3, where A 3 = 0\(A,UA 2). Moreover, A 3 i t s e l f has i measure 0 or ». In p a r t i c u l a r , i f feA and V ( x ) = {t9e0|f(0) = x}, then M ( V ( X ) n A ) = M ( A ) f o r a unique value of x £ 0, say x , unless 24 y(A ) = 0. Consequently, any f i n A can be w r i t t e n as x 1 « I ( A 1 ) i + x 2 * I ( A 2 ) , i . e . ( 0 ,M ) i s e q u i v a l e n t to (0 2,i>). • E v i d e n t l y , i n t a n g i b l e spaces are r a t h e r sparse. They are, however, of some i n t e r e s t and p r a c t i c a l importance, as the reader can judge from Examples 2.5.5 and 3.1.3, say. I f a space ( 0 ,M ) i s i n t a n g i b l e and i n c l u d e s two s e t s A,,A 2 with p r o p e r t i e s (I) and ( I I ) , we s h a l l say that i t i s dichotomous. With these d e f i n i t i o n s , we are now i n a p o s i t i o n to give a p r e c i s e statement of McConway's r e s u l t (the proof appears below): Theorem 2.2.4 (McConway 1981) Let (0,y) be a t a n g i b l e measurable space. A p o o l i n g operator n T:A —>A has the SSFP i f f n T ( f f ) = Z w f M - a . e . n i=1 i i n f o r some w £0, I w = 1 . i i = 1 i McConway observes that the r e s t r i c t i o n to what we c a l l t a n g i b l e spaces ( 0 ,M ) i s h a r d l y r e l e v a n t to h i s argument because i f 0 had only two neighbourhoods, no n o n t r i v i a l m a r g i n a l i z a t i o n c o u l d be performed. In view of Lemma 2.2.3 above, t h i s i s t r a n s p a r e n t . That Theorem 2.2.4 cannot be 25 g e n e r a l i z e d to i n t a n g i b l e spaces i s i l l u s t r a t e d by the f o l l o w i n g Example 2.2.5 Suppose 0 = {8yt62} and u = counting measure. Let G : [ 0 , 1 ] 2 — > [ 0 , 1 ] be such that G(x,y) = 0 i f x<y; x i f x=y; 1 i f x>y, and c o n s i d e r T:A 2—>A d e f i n e d by T ( f , , f 2 ) ( 0 ) = G ( f 1 ( 0 ) , f 2 ( 0 ) ) . Then T has the Strong Setwise F u n c t i o n Property and i t i s easy to check that there are no weights w, and w2 i n [0,1] f o r which T ( f , , f 2 ) = w,f, + w 2 f 2 f o r a l l f n , f 2 ' A. Proof of Theorem 2.2.4: One i m p l i c a t i o n i s obvious. To prove the other one, l e t A i , A 2 , A 3 be three neighbourhoods with p r o p e r t i e s (I) and ( I I ) , and c o n s i d e r 3 f = Z a ( j ) . I ( A ) i j=1 i j where a ( O ' / x f A , ) = x , a ( 2 ) « / u ( A 2 ) = y and a ( 3 ) « y ( A 3 ) = 1-x -i i i i i i y >0 f o r given x ,y i n [0,1] with x +y <1, 1<i<n. Then i i i i i (2.2.1) i m p l i e s J*I ( A , ) 'T( f f )dM = F ( x ) , n 26 ;i(A 2).T(£ 1 r...,f )dju = F ( y ) , n and a l s o / I ( A , U A 2 ) - T ( f , , . . . , f )du = F(x+y), n where x=(x,,...,x ), y = ( y 1 f . . . , y ) and x+y=(x,+y x +y ). n n n n T h e r e f o r e F s a t i s f i e s Cauchy's f u n c t i o n a l equation (2.1.1). n Using Lemma 2.1.2, i t f o l l o w s that F(x,,...,x ) = I w x on n i=1 i i n [0,1] f o r some w,,...,w in R. As F must be non-decreasing i n n each of i t s components, w > 0 f o r a l l i=1,...,n; moreover, the i n f a c t t h a t T ( f f ) i s always normalized f o r c e s I w = 1. • n i = 1 i At f i r s t s i g h t , MP seems s e n s i b l e . I t i s a p r i n c i p l e which guards a g a i n s t i n c o n s i s t e n c y of p r o b a b i l i t y assessments when performing a "marginal a n a l y s i s . " The necessary p o o l i n g can be accomplished e i t h e r before or a f t e r the marginals are r e p o r t e d ; the same r e s u l t i s obtained by e i t h e r route. However, there are two t y p i c a l s i t u a t i o n s i n which the need f o r a marginal a n a l y s i s w i l l a r i s e , and they both c a s t doubt on the v a l i d i t y of the MP p r i n c i p l e : Case I: 0 i s a product parameter space over which each member of a group of experts has been asked to produce h i s " m u l t i v a r i a t e " d i s t r i b u t i o n . However, the d e c i s i o n maker i s only i n t e r e s t e d i n 27 a p a r t i c u l a r v a r i a b l e . In that case, McConway e x p l a i n s that MP appears c o u n t e r - i n t u i t i v e , at l e a s t i f the a s s e s s o r s are experts i n d i f f e r i n g f i e l d s and hence have d i s p a r a t e p r i o r knowledge. I t c o u l d be, f o r i n s t a n c e , that only one a s s e s s o r has s p e c i a l i z e d knowledge in the v a r i a b l e of i n t e r e s t . Case I I ; 8 represents only one assessment s i t u a t i o n and each member of the panel has suggested h i s " u n i v a r i a t e " d i s t r i b u t i o n to the d e c i s i o n maker. In that case, when would the d e c i s i o n maker f e e l compelled to reduce the s i z e of 0 ( M ) , the c l a s s of a l l p o s s i b l e "events"? The answer i s : i n the l i g h t of new evidence r e v e a l e d to him (and to the panel) to the e f f e c t that c e r t a i n a l t e r n a t i v e s which had been c o n s i d e r e d p o s s i b l e a p r i o r i  can now be r u l e d out because they have become " a s s e r t e d l o g i c a l i m p o s s i b i l i t i e s " (Koopman 1940). Upon being a p p r i s e d of t h i s new i n f o r m a t i o n , he would update h i s b e l i e f s using Bayes' Theorem, the only r a t i o n a l p r e s c r i p t i o n f o r updating p r o b a b i l i t y d i s t r i b u t i o n s , s u b j e c t i v e or not. 1 Given that in general the processes of m a r g i n a l i z i n g and updating p r i o r s using Bayes' r u l e are not e q u i v a l e n t , what would seem to be c a l l e d f o r , here, i s not so much McConway's MP 1 In t h i s regard, c f . French (1982) who p r o v i d e s axioms j u s t i f y i n g the use of Bayes' Theorem when "changes of i n f o r m a t i o n take the form of the occurence of an event i n the f i e l d upon which the s u b j e c t i s c o n c e n t r a t i n g . " 28 condition as an axiom which would guarantee that the same f i n a l consensus d i s t r i b u t i o n i s arrived at whether ( i ) the experts' pr i o r s are combined f i r s t and the r e s u l t i n g consensus opinion i s updated; or else ( i i ) the posteriors are derived by each expert i n d i v i d u a l l y and then pooled by the decision maker. This axiom already e x i s t s , and Madansky (1964,1978) c a l l s i t External  Bayesianity. For a more technical d e f i n i t i o n of t h i s concept, as well as an analysis of some of i t s consequences, the reader i s referred to Section 2.4 below. Focusing on (2.2.1) now, a condition which i s e s s e n t i a l l y equivalent to MP, we see that the p r o b a b i l i t y assigned by the consensus d i s t r i b u t i o n , T(f ,,...,£ ), to any measurable set A e n fl(n) i s assumed to depend solely on the p r o b a b i l i t i e s given to A by the individual assessors' d i s t r i b u t i o n s f f . As n is p o t e n t i a l l y very large and A i s a r b i t r a r y in 0 ( M ) , t h i s condition i s obviously a far-reaching one. Indeed, not only does (2.2.1) dictate the l o c a l behaviour of T ( f , , . . . , f ) in n terms of the f 's (A's with 0 < M ( A ) << 1), i t also controls i t s i global behaviour (A's with M ( A ) >> 1) at the same time. We think that t h i s i s unnecessary and that i t should s u f f i c e to examine what T ( f f ) does on the atoms of n (for n d e f i n i t i o n , c f . Royden 1968, p. 321). Thus, in the following section, our f i r s t task w i l l consist of characterizing the linear opinion pool by assuming only what we have e a r l i e r 29 d e f i n e d as " l o c a l i t y " ( c f . D e f i n i t i o n 2.1.1), a c o n d i t i o n which amounts roughly to (2.2.1) r e s t r i c t e d to atoms. 2.3 A c h a r a c t e r i z a t i o n of the l i n e a r o p i n i o n pool v i a l o c a l i t y The purpose of t h i s s e c t i o n i s to e s t a b l i s h that the l i n e a r o p i n i o n pool i s the only l o c a l p o o l i n g operator which preserves dogmas. Both d e f i n i t i o n s were given i n S e c t i o n 2.1. The term " l o c a l i t y " mimics Bernardo (1979) who uses i t to d e s c r i b e an analogous property of u t i l i t y f u n c t i o n s . Roughly speaking, l o c a l i t y reduces p o o l i n g o p e r a t o r s to n Lebesgue-measurable f u n c t i o n s G on [0,°°) , and thus c o n s t i t u t e s a f a i r l y s trong requirement. But i t i s i n t u i t i v e l y a p p e a l i n g and c e r t a i n l y c o n s t i t u t e s a v i a b l e a l t e r n a t i v e to McConway's SSFP. I t c o u l d be viewed as a l i k e l i h o o d p r i n c i p l e f o r p o o l i n g o p e r a t o r s i n the f o l l o w i n g sense: a p a r t i c u l a r value of 8eQ i s c o r r e c t and the consensus p r o b a b i l i t y at 8 i s r e q u i r e d to depend only upon the p r o b a b i l i t i e s a s s i g n e d to the "true s t a t e , " and not upon the p r o b a b i l i t i e s of those s t a t e s of nature which c o u l d have ob t a i n e d but d i d not. The c o n d i t i o n that G be Lebesgue-measurable i s not needed e x p l i c i t l y here, but i t t i e s the present m a t e r i a l to that of the ensuing s e c t i o n s . We say that a p o o l i n g operator p r e s e r v e s dogmas i f i t d e f i n e s a consensus d e n s i t y which i s zero on that p a r t of the 30 space where the p r o b a b i l i t y a s s e s s o r s a l l s a i d that i t should be zero. The word "dogma" i s borrowed from Bacharach (1973) who observes that i f the p r o b a b i l i t y of a c e r t a i n event was assessed to be zero by an i n d i v i d u a l , no p e r t i n e n t evidence about that event w i l l ever a f f e c t h i s o p i n i o n . His judgement, s h a l l we say, r u l e s out i t s f o r c e i n advance, i . e . i t i s doqmatic. We do not wish to debate, at t h i s p o i n t , the problem of whether p r o f e s s i n g dogmas c l a s h e s with the p u t a t i v e l y s c i e n t i f i c a t t i t u d e of l e a v i n g a l l q u e s t i o n s open to be decided by " f a c t s . " However, i t seems reasonable to expect that a d e c i s i o n maker who has sought the advice of a r t i c u l a t e experts would not c h a l l e n g e t h e i r common dogmas. D i f f i c u l t i e s w i l l a r i s e only i f the dogmas expressed are c o n f l i c t i n g . Note that a l o c a l p o o l i n g operator need not always preserve n dogmas; c o n s i d e r f o r i n s t a n c e the operator T:A —>A which would map e v e r y t h i n g to I ( 0 ) / M ( 0 ) . However, as w i l l be shown i n the next lemma, dogma p r e s e r v a t i o n i s automatic when M ( 0 ) i s i n f i n i t e . Lemma 2.3.1 n Every l o c a l p o o l i n g operator T:A —>A pres e r v e s dogmas when M ( 0 ) i s i n f i n i t e . Proof; n Let G:[0,°°) —>[0,°°) be the Lebesgue-measurable f u n c t i o n whose 31 e x i s t e n c e i s guaranteed by D e f i n i t i o n 2.1.1. I f A e R ( M ) i s such that = K f o r some r e a l number 0<K<°°, l e t f = I ( A ) / K e A and observe that 1 = / T ( f , . . . , f ) d u = KG(1/K,...,1/K) + G( 0 , . . . , 0 ) •M ( 0 \ A ) = 1. Since G(o)-n(0\A) i s f i n i t e and u(Q\h) i s i n f i n i t e , we conclude that G(o) = 0, i . e . that T i s dogma p r e s e r v i n g . • We w i l l use methods from the theory of f u n c t i o n a l equations to prove: Theorem 2.3.2 Let ( 0 > r i ) be t a n g i b l e . The l i n e a r o p i n i o n pool i s the only l o c a l p o o l i n g operator which pre s e r v e s dogmas. We s p l i t the proof of Theorem 2.3.2 i n t o two lemmas. Lemma 2.3.3 Let A L F A 2 , A 3 be three ^-measurable neighbourhoods i n 0 with n p r o p e r t i e s (I) and ( I I ) . I f T:A —>A pr e s e r v e s dogmas and n there e x i s t s a Lebesgue-measurable f u n c t i o n G:[0,») —>[0,°°) such that T ( f , , . . . , f ) = G n ( f , , . . . , f ) y-a.e., n n n n then G(x) = I w x f o r a l l x e [0,1/M] , where M = min(M(A,), i = 1 i i M ( A 2 ) , u ( h 3 ) } . 32 Proof: C a l l m = M ( A ) , i = 1,2,3 and suppose that M = m3. There i s no i i l o s s of g e n e r a l i t y i n assuming A,,A 2,A 3 d i s j o i n t : i * j => A n A = i j 0. Consider 3 f = L a ( j ) - I ( A ) i j=1 i j 3 where a ( j ) £ 0 and Z a (j)«m = 1 f o r a l l 1<i<n. As T i j=1 i j p r e s e r v e s dogmas, G(o) = 0 and furthermore 3 / T ( f , , . . . , f )dM = I G( a ( j ) ) - m = 1 (2.3.1) n j=1 j where a ( j ) = ( a , ( j ) , . . . , a ( j ) ) , j = 1,2,3. n n Define h : [ 0 , l ] — > [ 0 , 1 ] by h(c) = 1 - M-G((1-c,)/M,...,(1-c )/M) n for a l l o ^ c < T , so that h(o) = 0 and h(T) = 1. It w i l l s u f f i c e to show that h s a t i s f i e s Cauchy's f u n c t i o n a l Equation (2.1.1), f o r then Lemma 2.1.2 w i l l imply that h(c) = n n I w c f o r some w ^ 0 , and I w = 1 because h(T) = 1. i = 1 i i i i = 1 i To e s t a b l i s h t h i s f a c t , note that by Equation (2.3.1), G l a d D - m , + G(a(2))-m 2 = h(c) (2.3.2) whenever a(3) = (T-c)/M. In p a r t i c u l a r , observe that i f o £ x < c ^ T are given, the c h o i c e s a (1) = x /m, and a (2) = (c -x )/m2 imply 33 Gfx/mJ.m, + G((c-x)/m 2).m 2 = h ( c ) . (2.3.3) However, t a k i n g a (1) = x /m, and a (2) = 0 i n (2.3.2) shows i i i that G(x/m,)'m, = h(x) and, s i m i l a r l y , choosing a (1) = 0 with a (2) = (c -x )/m2 i i i i e s t a b l i s h e s that G((c-x)/m 2)«m 2 = h ( c - x ) . Consequently, (2.3.3) becomes h(c) = h(x) + h(c-x) f o r a l l o -l x < c < i , which we can r e w r i t e as h(x+y) = h(x) + h(y) n n fo r x, y i n [ 0,1] with x+y e [ 0,1] (take y = c - x ) . T h i s concludes the p r o o f . • Lemma 2.3.4 Suppose that (0,M) i s t a n g i b l e , and l e t M 0 = i n f { j i ( A ) | A e J M M ) and 0 < M ( A ) < » ) } . Given 6 > 0 , i t i s always p o s s i b l e to f i n d three ^-measurable neighbourhoods A 1 R A 2 , A 3 which have p r o p e r t i e s (I) and (II) and are such that M 0 < M < M 0 +6, where M = m i n { M ( A , ) , M ( A 2 ) , M ( A 3 ) } . Proof; Since (0,M) i s t a n g i b l e , there e x i s t at l e a s t three u~ measurable neighbourhoods A , , A 2 , A 3 with p r o p e r t i e s (I) and ( I I ) . In f a c t , we might as w e l l assume that they are d i s j o i n t . Let 6 > 0 be given; we d i s t i n g u i s h two cas e s : 34 Case I; M 0 > 0 Choose B e fi(jx) to be such that M 0 ^ M ( B ) < M 0 + min{6,M 0} and look at B = Bn A , i = 1 ,2,3. Then M ( B )=0 or £ M 0 by the i i i d e f i n i t i o n of M 0, and there can be at most one i , say i=1, such 3 that M ( B ) > M 0, f o r 2M0 > M ( B ) > I u(B ). Our three s e t s are i i = 1 i B, , A 2 and A 3. Case I I ; M 0 = 0 Pick B e O ( M ) S O that 0 < M ( B ) < 6, and, once a g a i n , l e t B = i B n A , i =1 ,2,3. Put m = max{u(B^ ),u(B2),M(B3)} and r e l a b e l the i s e t s A ,B so that M ( B , ) = m. I f m=0, our c h o i c e of sets i s B, i i A 2,A 3; i f m>0, then r e p l a c e B by B,. • Theorem 2.3.2 i s an immediate consequence of the two lemmas above, and Example 2.2.5 shows that the t a n g i b i l i t y of the space ( 0 , M ) i s c r i t i c a l . Note a l s o t h a t , c o n t r a r y to what one might c o n j e c t u r e a t f i r s t , not every p o o l i n g operator i s l o c a l on an i n t a n g i b l e space. An example to t h i s e f f e c t f o l l o w s . Example 2.3.5 Let © = { 0 T , 0 2 } and M = counting measure, as i n Example 2.2.5 above. Let T:A 2—>A be d e f i n e d by T ( f , , f 2 ) ( 0 , ) = f , ( 6 , ) • f 2 ( 6 , ) and T( f , , f 2) (c92) = 1 ~ f i ( 0 i ) - f 2 ( 0 i ) . Then T pr e s e r v e s dogmas but i s not l o c a l . 35 I t i s p o s s i b l e to g e n e r a l i z e Theorem 2 . 3 . 2 i n at l e a s t one way. In the p r o p o s i t i o n s t a t e d below and proved i n the sequel, the d e f i n i t i o n of l o c a l i t y i s r e l a x e d t o allow the f u n c t i o n G to depend on 6 as w e l l as on the va l u e s that the d e n s i t i e s f f take at that p o i n t . Thus, we c o n s i d e r a l l p o o l i n g o p e r a t o r s n n T:A —>A of the form T ( f , , . . . , f ) ( e ) = G ( e , f , c e ) , . . . , f (e)) n-a.e. ( 2 . 3 . 4 ) n n n f o r some measurable f u n c t i o n G:0x[O,°>) — > [ 0 , ° ° ) . A p o o l i n g operator which s a t i s f i e s ( 2 . 3 . 4 ) w i l l be c a l l e d s e m i - l o c a l . Theorem 2 . 3 . 6 Let ( 6 , M ) be t a n g i b l e , and suppose that T i s a s e m i - l o c a l p o o l i n g operator which pre s e r v e s dogmas. If the G ( 0 , O of n Equation ( 2 . 3 . 4 ) i s continuous as a f u n c t i o n on [ 0 , = ° ) f o r u~ almost a l l 8 e ©, then T i s a l i n e a r o p i n i o n p o o l . Theorem 2 . 3 . 6 says that the c l a s s of s e m i - l o c a l n o n - l o c a l p o o l i n g o p e r a t o r s i s s m a l l . Indeed, when © i s coun t a b l e , |6| ^ 3 and M i s a counting measure, i t i s i n f a c t empty. For, take x n e [ 0 , 1 ] and l e t 6, 77, X be three elements i n 0 . I f f (e) = x = 1-f (X) and g (7?) = x = 1-g (X), the f a c t s that i i i i i i / G n ( f 1 , . . . , f )du = /Gn(g 1,...,g )dn = 1 and f (X) = g (X), n n i i 3 6 1 = 1 , . . . , n together e n t a i l G ( t 9 , x ) = G ( i ? , x ) . Our proof of Theorem 2 . 3 . 6 i s an attempt to g e n e r a l i z e t h i s argument to a r b i t r a r y spaces. I t i s c o n c e i v a b l e that the e x t r a ( c o n t i n u i t y ) c o n d i t i o n on G c o u l d be weakened, but we have not attempted to do so. Note that whatever be ( 0 , M ) , the requirement t h a t T pres e r v e s dogmas i s necessary to r u l e out p o o l i n g o p e r a t o r s which would map every n - t u p l e of o p i n i o n s (f f ) to the same f i x e d y - d e n s i t y g e n A . These o p e r a t o r s are worse than d i c t a t o r s h i p s , s i n c e they correspond to the case where the d e c i s i o n maker's mind i s made up i n advance and " c o n s u l t a t i o n " i s conducted f o r form's sake o n l y . They would t h e r e f o r e seem to be of l i t t l e i n t e r e s t . Here again, the r e s u l t f a i l s to extend to dichotomous spaces: Example 2 . 3 . 7 Let 0 = { 0 , , i 9 2 } , u = counting measure and c o n s i d e r a f u n c t i o n G : 0 X [ O , 1 ] 2 — > [ 0 , 1 ) d e f i n e d by G ( 0 , , x , y ) = x • I { y | 0 < y < 1 / 2 or y=l} + ( 1 -x ) • I {y | 0<y< 1 / 2 } and G ( t 9 2 , x , y ) = x - I { y | y = 0 or 1 / 2<y< 1 } + ( 1 - x ) - I { y | 0 < y < l / 2 } . Then T : A 2 — > A d e f i n e d by T ( f 1 , f 2 ) ( 0 ) G ( 6 , f , ( 6 ) , f 2 ( 6 ) ) i s s e m i - l o c a l and p r e s e r v e s dogmas. However, i t i s not l o c a l . The g i s t of the proof of Theorem 2 . 3 . 6 i s co n t a i n e d i n 37 Lemma 2.3.8 Let A , , A 2 , A 3 be three ^-measurable neighbourhoods i n 0 with n p r o p e r t i e s (I) and ( I I ) , and l e t T:A —>A be a p o o l i n g operator n which preserves dogmas. I f G:0x[O,°°) — > [ 0 , ° 0 i s a measurable f u n c t i o n such that (2.3.4) holds f o r a l l c h o i c e s of f f i n n n A, then there e x i s t w,,...,w e [0,1] with Z w = 1 f o r which n i = 1 i n G(«,x) = I w x M - a . e . on A (2.3.5) i=1 i i j n f o r a l l x e [ 0 , 1 /M ( A )] and j=1,2,3. j Proof: We d i v i d e the proof i n t o three p a r t s . n Step 1 ; Define f ( x ) = /I ( A , )G( • , x)d/i f o r a l l x e [ 0 , 1 / M ( A , ) ] ; we w i l l show that f s a t i s f i e s Cauchy's f u n c t i o n a l Equation (2.1.1). F i r s t note that i f f = x I ( A , ) + y I ( A 2 ) + z I ( A 3 ) i s i n A, i i i i ; T ( f , , . . . , f )dy = /I (A , )G( • ,x)di* + J l ( A 2 ) G ( • , y ) d y n + / I ( A 3 ) G ( • , z ) d M =1 (2.3.6) by the f a c t that T p r e s e r v e s dogmas. L e t t i n g g = y I ( A 2 ) + [1-y M ( A 2 ) ] • I ( A 3 ) / u ( h 3 ) and i i i h = y [ M ( A 2 ) / * I ( A 1 ) ] . I ( A 1 ) •+ [1-y M ( A 2 ) ] . I ( A 3 ) / M ( A 3 ) 3 8 both i n A,1^i<n, we see that /T(g,,...,g )&n = n ; i ( A 2 ) G ( - , y ) d M + ; i ( A 3 ) G ( . f [ T - y / * ( A 2 ) ] / M ( A 3 ) ) d M = 1 and a l s o /T(h,,...,h )dM = n / I ( A , ) G ( • ,yu(A2)/u(h,) )d/i + ; i ( A 3 ) G ( - , [ T - y M ( A 2 ) ] / M ( A 3 ) ) d M = 1. Thus J I ( A 2 ) G ( • , y ) d M = J I ( A 1 ) G ( ' f y M ( A 2 ) / M ( A 1 ) ) d i i = f ( y M ( A 2 ) / M ( A , ) ) . ( 2 . 3 . 7 ) S i m i l a r l y , we f i n d that / I ( A 3 ) G ( • ,z)d* i = / I ( A , ) G ( • , z n ( h 3 ) / n ( h , ) ) d M = f ( Z M ( A 3 ) / M ( A , ) ) ( 2 . 3 . 8 ) and so ( 2 . 3 . 6 ) now reads f ( x ) + f ( y M ( A 2 ) / j u ( A 1 ) ) + f ( Z M ( A 3 ) / M ( A 1 ) ) = 1 whenever X M ( A , ) + Y M ( A 2 ) + z / u ( A 3 ) = 1. R e l a b e l l i n g y = yn(h2)/M(A,), z = zn(A3)/n(A,), we have f ( x ) + n f ( y ) + f ( z ) = 1 f o r a l l x,y,z i n [ 0 , 1 / M ( A , ) ] with x + y + i i z = l / * i ( A , ) , 1<i<n. i n n So i f u, v are i n [ 0 , 1 / M ( A , ) ] with u + v e [ 0 , 1 / M ( A , ) ] , and i f z = ( 1 / M ( A , ) ) - U -v ,i=1,...,n, then f(u+v) + f ( o ) + f ( z ) = 1 39 and a l s o f (u) + f ( v ) + f(z) = 1. But again, f ( o) = 0 because T p r e s e r v e s dogmas, and so f(u+v) = f(u) + f ( v ) . A c c o r d i n g to Lemma 2 . 1 . 2 now, the r e e x i s t a,,...,a i n R such n n n that f ( x ) = L a x on [ 0 , 1 / M ( A , ) ] , and s i n c e f(x)> 0 always, i = 1 i i these constants a are non-negative. i n Furthermore, fCT/VtA,)) = 1 , so that Z a = A I ( A , ) . Just put i=1 i w = a /M(A,), 1<i<n. i i Step 2: We show that G(0,x) i s jtz -almost everywhere constant i n (9 on A=A,. For t h a t , we use the key f a c t that f o r any u~ measurable subset A' of A, n / I ( A ' ) G ( • , x ) d M = I w x M ( A ' ) . i = 1 i i If M ( A ' ) = 0 or = M ( A ) , t h i s i s obvious, and i f 0 < n ( A ' ) < / u ( A ) , we can apply the above argument to see that n / I ( A ' ) G ( • , x ) d M = L w'x M ( A ' ) i = 1 i i and n / I ( A " ) G ( • , x ) d M = Z w"x M ( A " ) , i=1 i i where A" = A\A' and the primes on the w 's i n d i c a t e a p o s s i b l e i dependence on the set over which G(«,x) i s i n t e g r a t e d . These primes may, i n f a c t , be dropped; f o r , i f 40 g = x I ( A ' ) + [ 1 - x M ( A ' ) ] - I ( A 2 ) / M ( A 2 ) i i i a n d h = x [ M ( A ' ) / M ( A " ) ] - I ( A " ) + [ 1 - x M ( A ' ) M ( A 2 ) / / i ( A 2 ) i i i a r e i n A , 1< i < n , t h e n /T(g,,...,g )d/x = / I ( A ' ) G ( • , x ) d M + J l ( A 2 ) G ( • , [ 1 -XM(A' ) ] / y ( A 2 ) )d / i n e q u a l s 1 a n d e q u a l s J T ( h 1 r . . . , h )du = ; I ( A " ) G ( . , X M ( A ' ) / * i ( A " ) ) d M n + / I ( A 2 ) G ( . , [ T - X M ( A ' ) ] / M ( A 2 ) ) d M , from which we conclude / I ( A ' ) G ( • , x ) d M = / I ( A " ) G ( • , X M ( A ' ) / y ( A " ) ) d M and i n t u r n n n Z w'x M(A') = Z w"x M(A') i=1 i i i=1 i i fo r every p o s s i b l e c h o i c e of x,,...,x i n [ 0 , 1//u(A) ]. Thus w' n i = w" f o r a l l i=1,...,n, and moreover w' = w s i n c e i i i / I ( A ) G ( • , x ) d M = J l ( A ' ) G ( • , x ) d M + / I ( A " ) G ( • f x ) d M e n t a i l s n n Z w x M(A) = Z w'x [ / i ( A ' ) + M ( A " ) ] , i = 1 i i i = 1 i i n n or Z w x Z w'x f o r a l l x,,...,x i n [ 0 , 1 / J U ( A ) ] . i=1 i i i=1 i i n n F i n a l l y , f i x x e [ 0 , 1 / M ( A ) ] and suppose that f o r some 6 > 0, n the set A' = {0eA|G(0,x) > Z w x +6} i s n o n - n e g l i g i b l e . Then i = 1 i i 41 n n / I ( A ' ) G ( • , x ) d M = Z w x n ( A ' ) > I w x M ( A ' ) + 5M ( A ' ) , i = 1 i i i = 1 i i n a c o n t r a d i c t i o n . Hence G(«,x) ^ Z w x n~a.e. on A, and a i = 1 i i s i m i l a r argument shows the reverse i n e q u a l i t y . Step 3; We can repeat steps 1 and 2 f o r A 2 or A 3 i n s t e a d of A,, so that n G(«,x) = Z w x ju-a.e. (2.3.9) i = 1 i j i n f o r a l l x e [0,1/M(A ) ] and some given c o n s t a n t s w i n [0,1] j i j n s a t i s f y i n g Z w = 1 , j =1,2,3. i=1 i j But by (2.3.7), n / I ( A 2 ) G ( • , y ) d M = Z w y M ( A 2 ) i=1 i2 i = / I ( A , ) G ( • , y M ( A 2 ) / M ( A , ) ) d y n Z w y u(A2) i=1 i1 i n f o r a l l y e [ 0 , 1 / M ( A 2 ) ] , so that w = w , i=1,...,n. i 1 i2 S i m i l a r l y , w = w f o l l o w s from (2.3.8) and so (2.3.9) e n t a i l s i 1 i3 the s t a t e d c o n c l u s i o n . • Let £ denote the c o l l e c t i o n 42 {A, eJ2(/n) IA! , A 2 ,A 3 have properties ( I ) & ( I I ) for some A 2 ,A 3 CJ2(M) } (note that £ * 0 <=> (0,M) i s t a n g i b l e ) . Lemma 2.3.8 can be strengthened i n the f o l l o w i n g way: Lemma 2.3.9 n Let (0,M) be t a n g i b l e and l e t T:A —>A be a dogma p r e s e r v i n g n s e m i - l o c a l p o o l i n g o p e r a t o r . I f G:0x[O,<=°) — > [ 0 , ° ° ) denotes the measurable f u n c t i o n f o r which (2.3.4) hol d s , then there e x i s t w, n ,...,w e [0,1] s a t i s f y i n g Z w = 1 and such that n i= 1 i n G ( • , x) = Z w x M~a.e. on A, i = 1 i i n whatever be x e [ 0 , 1 / M ( A ) 3 and A e £. Proof : n Let A and B i n £ be so that G(«,x) = Z w x M _a.e. on A i = 1 i i n and G(«,y) = Z w'y y-a.e. on B f o r some {w },{w'} i n [0,1] i=1 i i i i se t s of weights, each set adding up to 1, and a l l x i n n n [0,1//z(A)] , y i n [ 0,1/M (B)] . That the {w } and {w1 } e x i s t i i f o l l o w s from Lemma 2.3.8. If /i(AnB)> 0, then c l e a r l y w = w', i = 1,...,n. Otherwise, l e t i i 43 A, be a n o n - n e g l i g i b l e subset of A \ B and p i c k A 2 , A 3 e 0 ( M ) S O that A 1 f A 2 , A 3 have p r o p e r t i e s (I) and ( I I ) . Here, we used the f a c t t h a t (0,/z) i s t a n g i b l e . From Lemma 2.3.8 above, we know that n G(-,x) = I w x M-a.e. on A , (2.3.10) i=1 i i j n where x e [ 0 , 1 /M ( A )] and j =1,2,3. j If u(B n A ) > 0 f o r j=2 or 3, we are done, j If not, then { B , A 2 , A 3 } c o n s t i t u t e s a set with p r o p e r t i e s (I) and ( I I ) and we employ Lemma 2.3.8 again to conclude that n G(«,y) = Z w'y j i - a . e . on A , (2.3.11) i=1 i i j n y being a r b i t r a r y i n [0,1/M(A )] and j=2,3. P u l l i n g (2.3.10) j and (2.3.11) together shows that w' = w , i=1,...,n. • i i We say that a /u-density f e A i s a simple f u n c t i o n i f f f = Z c I ( A )fi-a.e. f o r some c £0 and a sequence { A efl( M) | i= 1 ,2. . .} i^1 i i i i of d i s j o i n t s e t s . With t h i s d e f i n i t i o n , we can s t a t e and prove P r o p o s i t i o n 2.3.10 n If T : A — > A i s a dogma p r e s e r v i n g s e m i - l o c a l p o o l i n g operator on a t a n g i b l e space (0,ju), then there e x i s t 6 [0,1] n n such t h a t Z w =1 and i = 1 i 44 n T ( f 1 f . . . , f ) = Z w f u-a.e. n i=1 i i fo r a l l f 1 f . . . , f simple f u n c t i o n s i n A. n Proof: If f f e A are simple f u n c t i o n s , i t i s p o s s i b l e to f i n d a n sequence S = { A e f i ( u ) | i = 1 , 2 , . . . } of d i s j o i n t s e t s A together i j with c o n s t a n t s O^c <=° f o r which f = Z c I ( A ) /n-a.e., i j i j2M i j j i=1,...,n. Since (Q,n) i s t a n g i b l e , we can assume that |S| > 2 , so t h a t the A 's belong to f . Thus, by Lemma 2 . 3 . 9 , there e x i s t j e [ 0 , 1 ] summing up to 1 f o r which ( 2 . 3 . 5 ) holds t r u e . n Since T i s s e m i - l o c a l , T ( f , , . . . , f ) = G(«,c ,...,c ) M - a . e . on n 1 j nj A f o r each j £ 1, and observe that c «M(A ) < Jf d/x ^  1 , so j i j j i n that G(',c ,...,c ) = Z w c M -a.e. on A by Lemma 2 . 3 . 9 . 1 j nj i=1 i i j j The f o r m u l a t i o n of the f o l l o w i n g c o r o l l a r y was suggested by Dr. Harry Joe (personal communication). C o r o l l a r y 2 . 3 . 1 1 I f ( 6 ,M) i s t a n g i b l e and u i s both o - f i n i t e and atomic, the l i n e a r o p i n i o n pool i s the only s e m i - l o c a l p o o l i n g operator 45 which p r e s e r v e s dogmas. Proof; If M i s a - f i n i t e and atomic, the c o l l e c t i o n C of i t s atoms i s at most countable; furthermore, |C| ^ 3 from the f a c t t h a t (0,M) i s t a n g i b l e . So, i f we w r i t e C = {A | i = 1 ,2, 3.. .}, i every f u n c t i o n f e A can be expressed ju -almost everywhere as an i n f i n i t e sum L c I(A ), i . e . A c o n s i s t s of simple f u n c t i o n s i*1 i i o n l y . Apply P r o p o s i t i o n 2.3.10. • In g e n e r a l , i t does not n e c e s s a r i l y f o l l o w from P r o p o s i t i o n 2.3.10 that any dogma p r e s e r v i n g s e m i - l o c a l p o o l i n g operator i s l o c a l . What i s c l e a r , however, i s that i f T i s continuous with resp e c t to the pointwise convergence topology, then Theorem 2.3.6 and P r o p o s i t i o n 2.3.10 are e q u i v a l e n t . T h i s r e g u l a r i t y c o n d i t i o n i s secured by r e q u i r i n g G i t s e l f to be continuous. Lemma 2.3.12 n n Let T:A —>A be s e m i - l o c a l and l e t G:0x[O,°°) —>[0,=°) be the c o r r e s p o n d i n g f u n c t i o n f o r which (2.3.4) h o l d s . I f G(6,-) i s n continuous as a f u n c t i o n on [0,°°) f o r M _ a l m o s t a l l 6 e 0, then Lim T ( f , .. . , f ) = T ( f f ) k—>°> 1k nk n whenever f — > f pointwise i x - a . e . as k—>», i = 1,...,n. ik i P roof: Let 46 n A = U { 0 e 0 | Lim f (0)#f (c9) } , i=1 k — > » ik i B = U { 0 e 0|T(f , . . . , f ) ( 0 ) * G ( 0,f ( 0 ) , . . . , f (6))}, k>1 1k nk 1k nk n and C = { 0 e 0|G ( 0 , « ) i s not continuous as a f u n c t i o n on [0,°°) }. Let a l s o D = { 0 e 0 | T ( f , , . . . , f ) ( 0 ) * G ( 0 , f , ( 0 ) , . . . , f ( 0 ) ) } and n n denote by E the y - n e g l i g i b l e set AUBUCUD. For a l l 0 e 0\E we have Lim T ( f , . . . , f ) ( 0 ) = Lim G ( 0,f ( 0 ) , . . . , f ( 0 ) ) k—><=° 1k nk k—>°° 1k nk = G ( 0 , f , ( 0 ) , . . . , f ( 0 ) ) n = T ( f , , . . . , f ) ( 0 ) , n i . e . Lim T ( f , . . . , f ) = T ( f 1 , . . . , f ) i n the pointwise k—>=° 1k nk n convergence topology. • To complete the proof of Theorem 2.3.6, i t s u f f i c e s to combine P r o p o s i t i o n 2.3.10 with the above lemma, keeping i n mind that every non-negative measurable f u n c t i o n on a space 0 i s the l i m i t of some sequence of simple f u n c t i o n s (Royde.n 1968, p. 224) . In c o n c l u s i o n , we have argued that when p o o l i n g o p i n i o n s on 0, a dominating measure n w i l l u s u a l l y impose i t s e l f as a n a t u r a l c h o i c e f o r both the experts and the d e c i s i o n maker. In that case, opi n i o n s take the form of d e n s i t i e s with res p e c t to M, and the c o n d i t i o n which we c a l l e d " l o c a l i t y " (or perhaps 47 s e m i - l o c a l i t y ) seems more r e a d i l y i n t e r p r e t a b l e than McConway's axiom (2.2.1). McConway 1s c h a r a c t e r i z a t i o n of the l i n e a r o p i n i o n p o o l can then be r e f o r m u l a t e d i n terms of l o c a l i t y and appears as Theorem 2.3.2. Theorem 2.3.6 extends t h i s r e s u l t to s o - c a l l e d " s e m i - l o c a l " p o o l i n g o p e r a t o r s . These f i n d i n g s w i l l have an important consequence i n the f o l l o w i n g s e c t i o n , where Madansky's idea of E x t e r n a l B a y e s i a n i t y w i l l be s t u d i e d at some l e n g t h . 2.4 Seeking E x t e r n a l l y Bayesian procedures In S e c t i o n 2.2, we suggested that E x t e r n a l B a y e s i a n i t y seemed a more a p p r o p r i a t e c r i t e r i o n f o r s e l e c t i n g p o o l i n g formulas than McConway's M a r g i n a l i z a t i o n P o s t u l a t e . In the present s e c t i o n , we w i l l give a p r e c i s e d e f i n i t i o n of t h i s concept and i n v e s t i g a t e some of i t s i m p l i c a t i o n s . In p a r t i c u l a r , c o n d i t i o n s w i l l be s t a t e d under which E x t e r n a l B a y e s i a n i t y c h a r a c t e r i z e s the l o g a r i t h m i c o p i n i o n pool (2.4.2). E x t e r n a l B a y e s i a n i t y (EB) has been in t r o d u c e d by Madansky (1964;1978) as an axiom of group r a t i o n a l i t y f o r s o l v i n g decision-making problems. The concept, however, can be r e a d i l y i n t e r p r e t e d w i t h i n our framework f o r the problem pf the panel of e x p e r t s . B a s i c a l l y , i f the panel were to use an E x t e r n a l l y Bayesian procedure to determine a consensus, they would be p e r c e i v e d as a c t i n g i n the manner of a s i n g l e Bayesian. T h i s 48 e n t a i l s updating t h e i r b e l i e f s i n accordance with Bayes' r u l e . To i n s u r e t h a t they would act i n a c o n s i s t e n t f a s h i o n , i t i s necessary t h a t the p o o l i n g procedure y i e l d the same r e s u l t whether they pool before or a f t e r updating t h e i r b e l i e f s i n the l i g h t of new i n f o r m a t i o n . More p r e c i s e l y , we have the f o l l o w i n g D e f i n i t i o n 2.4.1 n Let T:A —>A be a p o o l i n g o p e r a t o r . We say that T i s E x t e r n a l l y Bayesian i f f 0 < / * T ( f f )dn < », and n T [ * f , / j * f ,dM,...,*f / J * f djz] = n n * T ( f f ) / J # T ( f f ) d M M-a.e. (2.4.1) n n whenever * : 0 — > [ 0 , = ° ) i s a ^-measurable f u n c t i o n such that 0 < f<i>f dy < » f o r each 1<i<n (such a f u n c t i o n $ i s c a l l e d a i l i k e l i h o o d f u n c t i o n ) . Examples of E x t e r n a l l y Bayesian procedures are d i c t a t o r s h i p s and (provided i t i s w e l l d e f i n e d ) the l o g a r i t h m i c o p i n i o n p o o l , n w ( i ) n w ( i ) n T ( f , , . . . , f ) = n f // n f du, L w(i)=1. (2.4.2) n i=1 i i=1 i i=1 49 In h i s book on d e c i s i o n a n a l y s i s , R a i f f a (1968) i l l u s t r a t e s what can happen i f the processes of updating and p o o l i n g p r o b a b i l i t y d i s t r i b u t i o n s do not commute. He g i v e s an example (on a dichotomous space) in which two experts f i n d i t i n t h e i r own best i n t e r e s t to convince the d e c i s i o n maker to compute the consensus d i s t r i b u t i o n before he l e a r n s of the outcome of an experiment. They do so i n order to maximize the impact of t h e i r o p i n i o n s on the consensus p e r c e i v e d by the d e c i s i o n maker, r e g a r d l e s s of the outcome of the experiment. Such behaviour need not be e n t i r e l y s e l f i s h and motivated only by the d e s i r e "to win." In case they d i s a g r e e , i t would be q u i t e reasonable to expect that each expert would b e l i e v e he i s r i g h t . However, new, r e l e v a n t evidence should always be welcomed -by both the experts and the d e c i s i o n maker- and the q u e s t i o n of whether to update o p i n i o n s before or a f t e r a consensus i s found should not admit the p o s s i b i l i t y of the experts g a i n i n g some advantage f o r t h e i r o p i n i o n s over the new evidence by s t r a t e g i c manoeuvring. E x t e r n a l B a y e s i a n i t y has a l s o been advocated by Weerahandi & Zidek (1978) who c a l l i t " p r i o r - t o - p o s t e r i o r coherence." T h e i r r a t i o n a l e f o r u s i n g t h i s axiom d e r i v e s from the o b s e r v a t i o n that i f each expert i s a Bayesian, h i s p r i o r o p i n i o n might w e l l have been h i s p o s t e r i o r i n an e a r l i e r experiment, and that s i m i l a r l y , he w i l l use the p o s t e r i o r which w i l l r e s u l t from h i s present i n v e s t i g a t i o n s as h i s f u t u r e p r i o r . 50 Thus, a l l i n a l l , E x t e r n a l B a y e s i a n i t y seems to be an eminently reasonable p r e s c r i p t i o n f o r s e l e c t i n g "good" p o o l i n g o p e r a t o r s . We commence our a n a l y s i s of i t s i m p l i c a t i o n s with an easy lemma. Lemma 2.4.2 n Let T:A —>A be an E x t e r n a l l y Bayesian p o o l i n g o p e r a t o r . Then T p r e s e r v e s dogmas and furthermore T{t,,...,t ) = T(g,,...,g ) n n M -a.e. whenever f = g u~a.e. f o r a l l 1^i<n. i i Proof: Let f , , . . . , f e A be such that Z = { 6 e © | f A 6) = ... = f (0)=O} i s n n n o n - n e g l i g i b l e ( i . e . M ( Z ) > 0 ) . I f $ = I ( 0 \ Z ) , then * f = f and i i so /4>f du - 1 , 1<i<n. Using Equation (2.4.1), i t f o l l o w s that i J $ T ( f f )dM = K f o r some r e a l number 0<K<« and a l s o n T ( f f ) = $ T ( f f )/K n-a.e. n n But the righ t - h a n d s i d e equals 0 y-a.e. on Z , so that n S u p p ( T ( f f )) c u Supp(f ), i . e . T i s dogma p r e s e r v i n g . n i = 1 i To prove the second a s s e r t i o n , suppose that f = g y-a.e. and i i l e t A = {0e©|f (0) = g (6)} i n S2(M ) , i=1,2,...,n. Define A = i i i n U A and $ = 1 ( A ) . i = 1 i 51 Since uih) = 0, $f = f fi-a.e. and s i m i l a r l y #g = g jx-a.e. i i i i Consequently, ffcf d/n = f$g dju = 1 , and furthermore i i ; * T ( f f )dM = / * T ( g g )dM = 1 . n n And now, using the hypothesis that T i s E x t e r n a l l y Bayesian, we f i n d t h a t T ( * f , , . . . , * f ) = * . T ( f 1 r . . . r f ) Ai-a.e. n n and a l s o T(<t>g,, — ,<i>g ) = $'T(g ,,. .. ,g ) u~a.e. n n However, ^ f = 4>g everywhere, and hence i i T ( f 1 r . . . , f ) = T ( g 1 r . . . , g ) u~a.e. • n n S e c t i o n 2.2 above conveyed our view that new i n f o r m a t i o n w i l l more r e a l i s t i c a l l y cause a panel of experts to update t h e i r p r o b a b i l i t y d i s t r i b u t i o n s v i a Bayes' Theorem than to m a r g i n a l i z e them, although both procedures w i l l sometimes y i e l d the same answer. Had McConway p o s t u l a t e d E x t e r n a l B a y e s i a n i t y i n s t e a d of h i s M a r g i n a l i z a t i o n P o s t u l a t e , he would have obtained a very d i f f e r e n t r e s u l t : Theorem 2.4.3 (An I m p o s s i b i l i t y Theorem) Let (©,M) be t a n g i b l e . The only E x t e r n a l l y Bayesian l o c a l p o o l i n g o p e r a t o r s are d i c t a t o r s h i p s . 52 Proof: n Let T:A —>A be l o c a l and E x t e r n a l l y Bayesian. Then we know from Lemma 2.4.2 that T a l s o p r e s e r v e s dogmas, and hence n n T ( f , , . . . , f ) = I w «f /i-a.e. f o r some w £ 0, I w =1 as n i=1 i i i i=1 i a consequence of Theorem 2.3.2. We show that w = 1 f o r some j j = 1 ,..., n. Let A 1 f A 2 e S2(M) have p r o p e r t i e s (I) and ( I I ) . Such s e t s e x i s t because ( 0 , M ) i s t a n g i b l e , and we can take them to be d i s j o i n t . Now p i c k i * j i n {l,...,n} and c o n s i d e r f = I ( A , ) / n ( A , ) , f = i k I ( A 2 ) / M ( A 2 ) where k runs over the set of i n d i c e s {1,...,n}\{i}. If * = x-I (A, ) + y I ( A 2 ) f o r some x and y i n (0,°°), x*y, then Equation (2.4.1) a p p l i e d on A, i m p l i e s that w /x = w /[w «x + (1-w )«y]. i i i i Assuming t h a t w i s n e i t h e r 0 nor 1, we conclude that x=y, a i c o n t r a d i c t i o n . • Remark 2.4.4 Under the hypotheses of Theorem 2.3.6, the above r e s u l t a l s o holds fo r s e m i - l o c a l p o o l i n g o p e r a t o r s . Note that the c o n d i t i o n that (6,M) be t a n g i b l e i s i n d i s p e n s a b l e , as evidenced by the p o o l i n g operator of Example 2.2.5. T h i s theorem g e n e r a l i z e s Genest (1982) and c o n f l i c t s with a p r e v i o u s f i n d i n g of Weerahandi & Zidek (1978). In t h e i r 53 manuscript, these authors proposed a d e r i v a t i o n of the l o g a r i t h m i c o p i n i o n pool (2.4.2) based both on E x t e r n a l B a y e s i a n i t y and l o c a l i t y . In view of Theorem 2.4.3, t h i s i s only true when a l l w 's are 0 but one, and we i n t e r p r e t any i f u n c t i o n r a i s e d to the power 0 as the c h a r a c t e r i s t i c f u n c t i o n 1(0) of the whole space. We c a l l Theorem 2.4.3 an " I m p o s s i b i l i t y Theorem" to emphasize that d i c t a t o r s h i p s of o p i n i o n s cannot g e n e r a l l y be regarded as d e s i r a b l e . Indeed, we would be i n c l i n e d to f o l l o w Bacharach's (1975) p o l i c y on t h i s matter and make d i c t a t o r s h i p s i n a d m i s s i b l e . In t h i s case, the theorem would read: "there are no E x t e r n a l l y Bayesian l o c a l p o o l i n g o p e r a t o r s . " Next, we extend our search f o r E x t e r n a l l y Bayesian p o o l i n g o p e r a t o r s to the c l a s s of q u a s i - l o c a l procedures, i . e . o p e r a t o r s of the form T ( f , , . . . , f ) = n Gn(f f )//Gn(f ! , . . . ,f )dji /z-a.e. (2.4.3) n n n where G:[0,°°) —>[0,°°) i s a Lebesgue-measurable f u n c t i o n with the r a t h e r d i s t i n c t i v e p r o p e r t y that 54 0 < TGn( f f )dju < » n (2.4.4) f o r a l l c h o i c e s of f 1 r . . . , f i n A. T h i s d e f i n i t i o n of q u a s i -n l o c a l i t y i s e q u i v a l e n t to that given at the beginning of t h i s chapter ( D e f i n i t i o n 2.1.1). Note that G i s not unique, as we c o u l d m u l t i p l y top and bottom of the r i g h t - h a n d s i d e of (2.4.3) by any non-zero p o s i t i v e constant without a l t e r i n g T. We have a l r e a d y encountered one E x t e r n a l l y Bayesian q u a s i -l o c a l p o o l i n g operator, namely the l o g a r i t h m i c p o o l i n g formula n w ( i ) (2.4.2). Here, G(x) = IT x and i f *:0 — > [ 0 , » ) i s such i = 1 i that 0 < K = f<i>f du < =>, then i i n w ( i ) n w ( i ) T(*f,/K,,...,*f A ) = n [*f /K ] / ; n [#f /K ] du n n i= 1 i i i=1 i i n w ( i ) n w ( i ) = $ n f / ; * n f du i=1 i i=1 i = * T ( f , , . . . , f ) / J * T ( f , , . . . , f )du n n n p r o v i d e d Z w(i) = 1 . In order to ensure that C o n d i t i o n (2.4.4) i = 1 i s met, however, i t i s necessary to r e s t r i c t the domain of T to a s m a l l e r c l a s s of ^ - d e n s i t i e s f f f o r which the i n t e g r a l n n w ( i ) n w ( i ) JUt du i s s t r i c t l y p o s i t i v e ( t h a t J IT. f d(i i s always i=1 i i=1 i 55 f i n i t e f o l l o w s from Holder's i n e q u a l i t y , a t l e a s t when the w( i ) ' s are non-negative; c f . M a r s h a l l t O l k i n 1979, p. 457). Here, we have chosen to use A 0 = {feA|f*0 M-a.e.} both f o r s i m p l i c i t y and ease of e x p o s i t i o n . If A 0 * 0, our a n a l y s i s suggest a f a i r amount about the behaviour of E x t e r n a l l y n Bayesian q u a s i - l o c a l p o o l i n g o p e r a t o r s a c t i n g on A . In f a c t , knowing from Lemma 2.4.2 th a t q u a s i - l o c a l E x t e r n a l l y Bayesian procedures preserve dogmas, the only s i t u a t i o n which the r e s t r i c t i o n to A 0 f a i l s to encompass i s that where some event E in JHju) would have been deemed " i m p o s s i b l e " (zero p r o b a b i l i t y ) by some of the experts but not by a l l . That t h i s o c c a s i o n should a r i s e a f t e r the experts exchanged t h e i r views (as we have assumed they have) i s u n l i k e l y . Moreover, i t i s u n r e a l i s t i c to expect the d e c i s i o n maker to r e c o n c i l e the i r r e c o n c i l a b l e . T h i s somewhat p a t h o l o g i c a l s i t u a t i o n i s indeed not u n l i k e that faced in c o n v e n t i o n a l Bayesian a n a l y s i s when the p r i o r and l i k e l i h o o d f u n c t i o n s have d i s j o i n t support and some i m p r o v i s a t i o n i s c a l l e d f o r . The problem which we w i l l now address i s : are the r e any n E x t e r n a l l y Bayesian q u a s i - l o c a l p o o l i n g o p e r a t o r s T:A 0 — > A 0 besides (2.4.2)? The answer i s no, at l e a s t when one i s w i l l i n g 56 to make an e x t r a assumption about (0,M), namely Assumption 2.4.5 There e x i s t n o n - n e g l i g i b l e /n-measurable s e t s i n 6 of a r b i t r a r y small measure, i . e . V 6 e (0,») 3 A e 0(M) such that 0 < M(A) < 6. (2.4.5) Indeed, we w i l l now prove the f o l l o w i n g c h a r a c t e r i z a t i o n of the l o g a r i t h m i c p o o l i n g o p e r a t o r : Theorem 2.4.6 Suppose (0,ji) s a t i s f i e s Assumption 2.4.5. The l o g a r i t h m i c o p i n i o n pool (2.4.2) i s the only E x t e r n a l l y Bayesian q u a s i - l o c a l n p o o l i n g operator T:A 0 —>A 0. Remark 2.4.7 If (0,M) s a t i s f i e s Assumption 2.4.5, then c l e a r l y i t i s t a n g i b l e and © i s i n f i n i t e . Thus Theorem 2.4.6 above does not cover the important case where © i s f i n i t e . The answer i n the l a t t e r case i s unknown. A s p e c i a l case of the f o l l o w i n g lemma w i l l prove u s e f u l i n e s t a b l i s h i n g Theorem 2.4.6: Lemma 2.4.8 Suppose (©,M) s a t i s f i e s Assumption 2.4.5. Given 5 > 0, there e x i s t s a sequence (A efi(M)|n=1,2,...} of mutually d i s j o i n t s e t s n 57 such t h a t 0 < M(A ) < 5 f o r a l l n £ 1. n Proof; The proof i s by i n d u c t i o n . I f 5 > 0 i s given and A,,...,A are n n mutually d i s j o i n t /i-measurable neighbourhoods with 0 < M(A ) i < 6, i=1,...,n, l e t B e Sl(u) such that 0 < M(B) < (1/2)-min{M(A )|l<i<n} < 6. i Then M ( B ) < 8, M ( A \ B ) £ M ( A )/2 and so { A , \ B , . . . , A \ B , B ] i i n forms a c o l l e c t i o n of n+1 mutually d i s j o i n t s e t s i n . • Another obvious consequence of Assumption 2.4.5 i s that the n f u n c t i o n G i n (2.4.3) must be d e f i n e d everywhere on (0,°°) : Lemma 2.4.9 Suppose (©,M) s a t i s f i e s Assumption 2.4.5, and l e t x,,...,x be given i n (0,<=°). There e x i s t f f i n A 0 such n n n that M( n {0e6|f (0)=x }) > 0. i = 1 i i Proof: Write 6 = min{l/x |l<i<n} and use Lemma 2.4.8 to choose A e i n(/x) such that 0 < M(A) < 8 and u(Q\k) > 0. I f h e A 0 i s any given jx-density, then JhI (0\A)dM > 0 f o r otherwise h would 58 v a n i s h on some set of s t r i c t l y p o s i t i v e measure, a c o n t r a d i c t i o n . Def ine f = x - I ( A ) + p - h i ( 0 \ A ) i i i where p = [1-x M ( A ) ] / f h I ( 0 \ A ) d M , i=1,...,n. C l e a r l y f e A 0 i i i n and n (0e6|f (0)=x } = A i s n o n - n e g l i g i b l e . • i=1 i i We s t a r t the proof of Theorem 2.4.6 with P r o p o s i t i o n 2.4.10 n Suppose (0,At) s a t i s f i e s Assumption 2.4.5. If T:A 0 — > A 0 i s E x t e r n a l l y Bayesian and of the form (2.4.3) f o r some Lebesgue-n measurable G:(0,=>) — > ( 0 , » ) , then G(cx) = c G ( x ) f o r a l l c^O n and x e (0,°°) . Proof: If c=0, then G(o)=0 by Lemma 2.4.2 (T p r e s e r v e s dogmas). So suppose c>0 and l e t x>o be f i x e d . -1 Given 6 = min{[x (c+1)] |l<i<n}, we can use Lemma 2.4.8 to f i n d i f i v e d i s j o i n t elements A , B , C , D , E of 0(M) with measure i n (0,6). Let 7 > 0 be such that -1 7 < min{2[x - J » ( A ) - C M ( B ) ] / M ( A U B ) } i 59 and p i c k 0 < X,i; < » so that -1 X < min{-7+2^[x ~u(A)-cu(B) ] / u ( A U B ) | 1 <Un} i -1 < max{-7+2-[x -M(A)-CM(B)]/M(AUB)|1<i<n} < £. i Now f o r each 1<i<n, there e x i s t s d e (0,1) so that i -1 Xd + £(1-d ) = - 7 + 2»[x - M ( A ) - C M ( B ) ] / M ( A U B ) . i i' i Defi n e f as i I ( A U B ) / 2 M ( A U B ) + d . I ( C ) / 4 M ( C ) + (1-d ) - I ( D ) / 4 M ( D ) + h-l(N)/4S, i i where N = 0\(AUBUCUD) , S = /hI(N)dii, and h i s some a r b i t r a r y f u n c t i o n i n A 0. Note t h a t , here again, J*h»I(N)dM > 0 f o r otherwise h would v a n i s h on E, a set of s t r i c t l y p o s i t i v e measure. I t i s easy to check that f belongs to A 0 . i Now c o n s i d e r $ 1 ( A ) + c I ( B ) + X I ( C ) + £I(D) + 7 K N ) . We have that $-f t 0 /u-a.e. and i J * - f du = [M(A)+CM(B)]/2M(AUB) + Xd /4 + $(\-d )/4 + 7/4 i i i -1 ... = [2x M ( A U B ) ] = K . i i Write u = l / 2 i i ( A U B ) , so that u = x K , 1<i<n. Now T i s i i E x t e r n a l l y Bayesian, i . e . 60 G ( * f , / R , , . . . , ^ /K ) J G ( * f , / K l f . . . , # f /K )dM (2.4.6) n n = n n #G(f,,. .. ,f ) / * G ( f i , . . . , f )du n n Observe that the r i g h t - h a n d s i d e of t h i s e x p r e s s i o n i s a constant independent of the set (A,B,C,D or N) on which both # and the f 's are e v a l u a t e d . So, i n p a r t i c u l a r , the l e f t - h a n d i s i d e i s the same whether on A or on B. Hence G(u/K,,...,u/K ) = (1/c)-Gtcu/K,,...,cu/K ) n n upon c a n c e l l i n g a common f a c t o r of G(u,...,u) on both s i d e s of the e q u a t i o n . R e c a l l i n g the d e f i n i t i o n of u and of the K 's, we i f i n d that c*G(x 1 f...,x ) = G(cx,,...,cx ), n n as a s s e r t e d i n the statement of the p r o p o s i t i o n . • n Thus i f a p o o l i n g operator T:A 0 — > A 0 i s both q u a s i - l o c a l and E x t e r n a l l y Bayesian, P r o p o s i t i o n 2.4.10 above t e l l s us t hat i t s c o rresponding G must be at l e a s t "homogeneous." However, the technique which we have used to reach t h i s c o n c l u s i o n c o u l d not be a p p l i e d s u c c e s s f u l l y i n cases when G need not be d e f i n e d n over the e n t i r e t y of (0,°°) , as when © i s f i n i t e f o r example. Not a l l homogeneous G's generate an E x t e r n a l l y Bayesian q u a s i - l o c a l p o o l i n g o p e r a t o r . Consider f o r i n s t a n c e the f u n c t i o n G(x) = max{x |l<i<n}, which g i v e s r i s e to the q u a s i -i 61 l o c a l procedure T ( f f ) = max{f f } / / m a x { f f }du. n n n C l e a r l y G i s homogeneous, but T i s not E x t e r n a l l y Bayesian, as P r o p o s i t i o n 2.4.11 w i l l now e s t a b l i s h . P r o p o s i t i o n 2.4.11 Let T be a v e c t o r of ones and w r i t e S>y f o r the v e c t o r (x,y, ,...,x y ). Then G(x)«G(y) = G(x«y)«G(T) f o r a l l x,y v e c t o r s n n n i n (0,°°) , where G i s the f u n c t i o n s p e c i f i e d in Equation (2.4.3). Proof: Let -1 0 < 7 < min{l,x |i<i<n}, i -1 0 < 6 < m i n { ( 1 ~ 7 ) / y ,(x -y)/y }, i i i and l e t A,B,C,D be d i s j o i n t elements of J2(M) with measure i n (0,6). If we w r i t e t = 1-(y+y u(B))r then t > 0 and 1/x > i i i i 7 + y M(B) f o r a l l i = 1 , . . . , n . i Next, choose 0 < X,£ < » so that -1 -1 X < min{t «[x - 7 - y M(B)]|l<i<n} i i i - 1 - 1 < max{t -[x - 7 - y n(B)]|1<i<n} < £. 62 Then f o r each 1<i<n there e x i s t s a unique d e ( 0 , 1 ) such that i -1 -1 Xd + £(1-d ) = t [x -7-y u(B)]. i i i i i D efine f = 7 I ( A ) / 2 M ( A ) + y 1 ( B ) + t d I ( C ) / M ( C ) i i i i + t ( 1 - d ) I ( D ) / M ( D ) + ( 7 / 2 ) • [ h i ( N ) / / h I ( N ) d M ] i i where h e A 0 i s a r b i t r a r y , and N = e\(AUBUCUD). (That ThI(N)dM i s not 0 i s a consequence of Lemma 2.4.8.) Now f * 0 i i - a . e . and i ft du = 7 + y j u + t = 1 , i i i and hence f e A 0 , 1 ^ i ^ n . i C o nsider $ = I(AUB) + I(N) + XI(C) + £I(D); we have / * f du = 7/2 + y ju(B) + t d X + t ( 1 - d )$ + 7/2 i i i i i i = 1/x f o r 1<i£n, i and s i n c e <I>f ^ 0 u~a.e., we may use the f a c t t h at T i s i E x t e r n a l l y Bayesian to deduce that the l e f t - h a n d s i d e of Equation ( 2 . 4 . 6 ) remains constant as the f 's and $ are i e v a l u a t e d on A and B r e s p e c t i v e l y . Consequently, we f i n d G( |3x |3x ) G(x 1y,,...,x y ) i n = ^ n n G(/3, .. . ,/3) G(y ! , . . . ,y ) n where /3 = 7/2M(A). But by P r o p o s i t i o n 2.4.10, the l e f t - h a n d 63 s i d e reduces to G(x 1 f...,x )/G(1,1,...,1), whence the r e s u l t . • n Proof of Theorem 2.4.6: n Consider H(x) = G(x)/G(T), a f u n c t i o n d e f i n e d on (0,<=°) . Then H i s Lebesgue-measurable and i t f o l l o w s from P r o p o s i t i o n 2.4.11 that H(x'y) = H(x)«H(y) on i t s domain. By Lemma 2.1.3, we conclude to the e x i s t e n c e of n r e a l numbers w(1),...,w(n) such n w ( i ) that H(x) = n x always, i . e . i = 1 i n w ( i ) G(x) = G(T)• n x i = 1 i There f o r e n w ( i ) n w ( i ) T ( f 1 f . . . , f ) = FI f // n f du n-a.e. n i=1 i i= 1 i n The f a c t that L w(i) = 1 f o l l o w s d i r e c t l y from P r o p o s i t i o n i = 1 2.4.10: i f x>0 and c>0 are given, we have Zw(i) n w(i ) G(cx)/G(T) = c • n x i=1 i Iw(i) = c -G(x)/G(T) = cG(x)/G(T). T h i s completes the proof of Theorem 2.4.6. • Summarizing our i n v e s t i g a t i o n s on q u a s i - l o c a l i t y and E x t e r n a l B a y e s i a n i t y , we have seen that p r o v i d e d the c l a s s of  " a d m i s s i b l e " ^ - d e n s i t i e s i s s u i t a b l y r e s t r i c t e d : 64 n w ( i ) ( i ) the l o g a r i t h m i c o p i n i o n pool II f i = 1 i n w ( i ) / ; n f i = 1 i d/u with n Z w(i) = 1 i s E x t e r n a l l y Bayesian whatever (@,u); i = 1 ( i i ) i f (0,y) s a t i s f i e s Assumption 2.4.5, the l o g a r i t h m i c o p i n i o n pool i s the only E x t e r n a l l y Bayesian q u a s i - l o c a l p o o l i n g procedure a v a i l a b l e . In f a c t , t h i s second c o n c l u s i o n can be somewhat strengthened, as we w i l l p r e s e n t l y show: P r o p o s i t i o n 2.4.12 If (0,M) i s such that £l(u) c o n t a i n s an i n f i n i t e sequence {A | n>1} of mutually d i s j o i n t s e t s , then the l o g a r i t h m i c n o p i n i o n p o o l i n g operator (2.4.2) i s not q u a s i - l o c a l unless w,,...,w are taken to be non-negative. Proof : I t s u f f i c e s to show that given a > 0, we can f i n d f,g e A 0 with J*f'(f/g) dM = 0 0. For, i f w <0 f o r some i e { 1 , 2 , . . . , n} , l e t a -w >0 and c o n s i d e r f = g, f = f , j * i , so that / f . ( f / g ) du -i i j », a c o n t r a d i c t i o n . Use Lemma 2.4.8 to f i n d a sequence {A |n£l} of d i s j o i n t u~ n a l a n 65 measurable neighbourhoods, and define f = Z K ! I ( A ) / [ M ( A ) i 2 ] + K 2hl(N)//hI(N)dy i£1 i i for some h e A 0 . (If fhI(N)dy = 0, then N = 0 \ ( U A ) has i£1 i measure zero, in which case l e t £ = R, I I ( A ) / [ M ( A ) i 2 ] i£1 i i instead.) In order that f be in A 0 , i t i s necessary to have K 1 T T 2 / 6 + K 2 = 1, so K i , K 2 > 0 can be chosen accordingly. c Put g = L,« Z I ( A )/n(k ) i + L 2hl(N)/fhI(N)dn where c equals i*1 i i c 2(a+l)/a > 2. Then g £ 0 M-a.e. and J*gd/i = L, Z 1/i + L 2 can be made equal to 1 with appropriate choices of L,,L 2, since c 0 < Z 1/i < OD. i£1 a a+1 c a Now Jf . ( f / g ) dM ^ Z [ K , / M ( A ) i 2 ] - U ( A ) i /L,] *x(A ) i£l i i i a+1 a K , • Z 1 = OD. • i£1 In t h i s last proposition, the hypothesis that there be at least countably many d i s j o i n t neighbourhoods in 6 i s c l e a r l y necessary. If (0,M) i s f i n i t e , there i s no reason why some of n the w 's could not be s t r i c t l y negative, as long as Z w =1. We complete t h i s section with an example to show that an 66 Externally Bayesian operator need not always preserve unanimity. Example 2.4.13 Let (0,xi) be dichotomous or tangible, so that there exist A,,A2 e Qiu) with properties (I) and ( I I ) . Write A = A, and B = ©\A, so that min{/j(A) , M ( B ) } > 0. Next, define g = 1(A) + I(B)/2 and n l e t T:A —>A be defined by T ( f 1 f . . . , f ) = ftg/JftgdM n (note that 1/2 < /f,gdM < 1 since 1/2 £ g £ 1). Obviously T does not preserve unanimity and i s nevertheless Externally Bayesian. However, i t i s neither loca, nor semi-local, nor even qu a s i - l o c a l ! 2.5 Information maximizing and divergence minimizing  pooling operators In t h i s section, we take a d i f f e r e n t approach to the problem of adequately describing a consensus of opinions. In the f i r s t part, we adopt the point of view that each opinion f i contains some "information" about © and we look for a single representative p r o b a b i l i t y d i s t r i b u t i o n , T ( f f ), whose n expected information content w i l l be a maximum. The pooling formula so obtained w i l l d i f f e r according to which d e f i n i t i o n of information i s elected. This approach w i l l be seen to have the 67 merit of providing a sensible interpretation of the constants w i with which each opinion f i s weighted, a question which was i l e f t unanswered by our previous attempts. Then, in the second part, we employ Kullback's (1968) concept of divergence between pr o b a b i l i t y d i s t r i b u t i o n s to construct a class of pooling formulas which contains both the linear and the logarithmic pools as l i m i t i n g cases. We begin with a short review of Shannon's d e f i n i t i o n of entropy, which i s basic to the Theory of Information. For convenience, we work on A 0, defined in Section 2.4 to be {feA|f£0 M~a.e.}. Perhaps the most celebrated and popular measure of the amount of information contained in a p r o b a b i l i t y density f on 0 is the entropy function E(f) = - J f - l o g ( f )d*x e [0,»), (2.5.1) the discrete version of which was introduced by Shannon (1948) in the context of communication engineering. The quantity E(f) measures the "uncertainty" contained in the random variable 0 (as governed by f) and thus represents, in some sense, our best knowledge of 0. The smaller the entropy, the less uncertain i s 0 and therefore the better informed one i s deemed to be upon being apprised of f. Strong j u s t i f i c a t i o n for using (2.5.1) has been supplied by way of axiomatic characterizations, though only in the discrete case. Most derivations, including those of 68 Faddeev (1956) and Forte (1973), are based on some v e r s i o n of the a d d i t i v i t y p o s t u l a t e which s t i p u l a t e s that the information expected from two experiments equals the information expected from the f i r s t experiment plus the c o n d i t i o n a l information (entropy) of the second experiment with respect to the f i r s t . This p o s t u l a t e must be considered fundamental to any idea of "informat i o n . " The f o l l o w i n g expression f o r the entropy of one p r o b a b i l i t y d ensity f i n A 0 with respect to another p r o b a b i l i t y d ensity g i s u s u a l l y known as the K u l l b a c k - L e i b l e r Information f o r d i s c r i m i n a t i n g between f and g: I ( f , g ) = / f . l o g ( g / f ) d n . (2.5.2) I t was d e f i n e d by Shannon (1948) i n the d i s c r e t e case and l a t e r extended by Kullback & L e i b l e r (1951) to the general case. The quotient l o g [ g ( 6 ) / f ( 6 ) ] may be i n t e r p r e t e d as the "weight of evidence" (Good 1950) or the information i n © = 6 for d i s c r i m i n a t i n g i n favour of H,: "the true d i s t r i b u t i o n i s g" versus H 0: "the true d i s t r i b u t i o n i s f." A l t e r n a t e l y , the q u a n t i t y (2.5.2) may be regarded as the information gain (a negative q u a n t i t y here) E( f ) - [-/f .log(g)dix] i n c u r r e d by using one's "best knowledge of ©," g, to take 69 d e c i s i o n s , while the true ( h y p o t h e t i c a l ) u n d e r l y i n g p r o b a b i l i t y d i s t r i b u t i o n governing 0 i s f . Two b a s i c p r o p e r t i e s of the K u l l b a c k - L e i b l e r i n f o r m a t i o n are embodied i n Lemma 2.5.1 Let I : ( A 0 ) 2 — > R be d e f i n e d by Equation (2.5.2). If f and g represent any non-vanishing ^ - d e n s i t i e s i n A 0, then ( i ) I ( f , g ) £ 0 always, and ( i i ) I ( f , g ) = 0 i f f f=g *i-a.e. Proof: T h i s r e s u l t i s s t a t e d and proved as Theorem 3.1 i n Chapter 2 of Kul l b a c k (1968). • The above p r o p e r t i e s of the K u l l b a c k - L e i b l e r Information measure are enough to suggest a new c h a r a c t e r i z a t i o n of the l i n e a r o p i n i o n pool i n the f o l l o w i n g c o n t e x t . Let us imagine f o r a moment that a d e c i s i o n maker has c o l l e c t e d n expert p r o b a b i l i t y assessments f f about 0 and that he i s n informed, knows or judges somehow that ( i ) one of these i s the d e n s i t y of the " o b j e c t i v e " p r o b a b i l i t y d i s t r i b u t i o n governing 0 as a random v a r i a b l e ; and ( i i ) the p r o b a b i l i t y that the i - t h n d i s t r i b u t i o n , f , i s o b j e c t i v e i s p £ 0, I p = 1. We have i i i=1 i a l r e a d y remarked i n Chapter 1 t h a t an o b j e c t i v e d i s t r i b u t i o n f o r 70 0 may only be v i r t u a l (© may be observable only once, for i n s t a n c e ) , so the s i t u a t i o n which we are d e s c r i b i n g i s h y p o t h e t i c a l . However, i t i s suggestive and d e s c r i p t i v e . I f f were the d e n s i t y of the o b j e c t i v e d i s t r i b u t i o n , then i a ccording to (2.5.2), the amount of information l o s t due to adopting a p r o b a b i l i t y d i s t r i b u t i o n g i n s t e a d of f would be i - I ( f ,g) = Jf - l o g ( f /g)dy. i i i Averaging over the f 's, we f i n d that the g l o b a l expected i n f o r -i mation l o s s i s n - L p - I ( f ,g), (2.5.3) i=1 i i a f u n c t i o n a l depending s o l e l y on g. I t would seem n a t u r a l to choose g so as to minimize (2.5.3), i . e . p i c k a p r o b a b i l i t y d i s t r i b u t i o n which minimizes the expected l o s s of information occasioned by the need to compromise. Note that the d e f i n i t i o n of p o o l i n g operator r u l e s out the p o s s i b i l i t y that g could be randomly chosen from the f 's: although a t t r a c t i v e , t h i s i s e l e c t i o n scheme does not engender the idea of consensus. I f a n p o o l i n g operator T:A0 —>A 0 i s such that T ( f f ) = g n 71 minimizes (2.5.3) whatever be f , , . . . , f ( ( p 1 r . . . , p ) being a n n f i x e d v e c t o r of p r o b a b i l i t i e s ) , we say that i t i s a K u l l b a c k - L e i b l e r Information Maximizer (KLIM). Theorem 2.5.2 The l i n e a r o p i n i o n pool T ( f f ) n moreover, w = p , i=1,...,n. i i Proof; n C a l l f = Z p f . To be a KLIM, g must minimize (2.5.3) or, i = 1 i i e q u i v a l e n t l y , maximize n Z p / f -log(g/f)dM = I ( f , g ) • i = l i i Lemma 2.5.1 shows that g = f j i-a.e. • Here, we have a c h a r a c t e r i z a t i o n of the l i n e a r o p i n i o n pool which does not impose a s p e c i f i c form on the p o o l i n g operator at the o u t s e t . L o c a l i t y merely comes as a consequence of the d e f i n i t i o n of T. A l s o noteworthy i s the f a c t that t h i s r e s u l t does not d i s t i n g u i s h between t a n g i b l e and i n t a n g i b l e spaces. Theorem 2.5.2 p r o v i d e s us with a n a t u r a l i n t e r p r e t a t i o n of the weights, w , at l e a s t as they appear i n the l i n e a r o p i n i o n i p o o l . If an o b j e c t i v e p r o b a b i l i t y d e n s i t y , f, f o r 8 and n = Z w f i s the only KLIM; i=1 i i 72 o b j e c t i v e p r o b a b i l i t i e s , p , of {f =f} e x i s t , we have seen that i i w should equal p , 1<i<n. When f e x i s t s but the p 's are i i i unknown, i t would seem n a t u r a l to l e t w represent the d e c i s i o n i maker's s u b j e c t i v e p r o b a b i l i t y t h a t the i - t h expert o p i n i o n i s the " r i g h t one." T h i s supports the i n t u i t i v e idea that even i n the absence of an o b j e c t i v e d i s t r i b u t i o n , f , the weights, w , i s hould be chosen on the b a s i s of a s u b j e c t i v e judgement made by the d e c i s i o n maker concerning the accuracy of each a s s e s s o r . Winkler (1968) d e s c r i b e s some of the most popular r u l e s f o r determining the weights. A l l of them are based on the i n t u i t i v e grounds proposed above. The most promising one, suggested by Roberts (1965), looks at l i k e l i h o o d r a t i o s to compare the p r e d i c t i v e a b i l i t y of the e x p e r t s ; t h i s i n v o l v e s the a p p l i c a t i o n of Bayes' Theorem to f o r m a l l y r e v i s e the weights a f t e r each assessment and the r e l a t e d o b s e r v a t i o n . More simply, though, the d e c i s i o n maker c o u l d use the present methods to e x t r a c t a consensus on the weights a f t e r having asked each expert, i , t o produce a set of weights {w |i£j<n} on the b a s i s of the i j r e l a t i v e importance that he would a s s i g n t o the o p i n i o n s of the v a r i o u s members of the panel, i n c l u d i n g h i m s e l f . Of course, t h i s r a i s e s f u r t h e r q u e s t i o n s about the formula to be used i n p o o l i n g the weights and the value (weight) t o be assigned to any p a r t i c u l a r weights assessment. In p r i n c i p l e , t h i s process c o u l d go on f o r ever, except that the f i n a l consensus w i l l g e n e r a l l y 73 be l e s s s e n s i t i v e to the c h o i c e of weights than to the c h o i c e of the p o o l i n g formula. "In a way,' s t a t e M o s t e l l e r & Wallace (1964, p. 264), ' t h i s i s an o l d s t o r y i n s t a t i s t i c s because modest changes i n weights o r d i n a r i l y change the output modestly." The idea of maximizing the expected K u l l b a c k - L e i b l e r i n f o r m a t i o n i s not new. I t has been suggested by L i n d l e y (1956) as a s e n s i b l e (but ad hoc) c r i t e r i o n f o r experimental designs "where the o b j e c t of experimentation i s not to reach d e c i s i o n s but r a t h e r to gain knowledge about the world." I t i s p r e c i s e l y the c o n t e x t i n which t h i s idea has been a p p l i e d here: c o l l e c t i n g expert o p i n i o n s may be viewed as an experiment, and i t i s the s t a t e d purpose of our problem to assess the r e l a t i v e l i k e l i h o o d of the v a r i o u s p o s s i b l e s t a t e s of nature, not to take d e c i s i o n s . Bernardo (1979) has shown that t h i s maximization procedure i s but another instance of the general (Bayesian) p r i n c i p l e of maximizing the expected u t i l i t y , and that i t i s , i n some sense, the o n l y s e n s i b l e one when the o b j e c t i s to make i n f e r e n c e without any s p e c i f i c a p p l i c a t i o n i n mind. In Chapter 1, we drew a p a r a l l e l between the problem of determining a consensus and that of e s t i m a t i n g a q u a n t i t y from a number of o b s e r v a t i o n s . We have seen that the l i n e a r o p i n i o n pool may be i n t e r p r e t e d as a kind of s u f f i c i e n t s t a t i s t i c , a s t a t i s t i c which, a c c o r d i n g to F i s h e r (1934), "summarises the whole of the r e l e v a n t i n f o r m a t i o n s u p p l i e d by the sample." 74 I t would seem n a t u r a l t o t r y to extend Theorem 2.5.2 to the s o - c a l l e d Renyi Information measures -1 a 1 -a I ( f ,g) = (1-a) . log[Jf g dii], 0<a<1 (2.5.4) a based on the g-entropy f u n c t i o n s -1 a E ( f ) » (1-a) -log [J f dn], 0<a<1 a i n t r o d u c e d by Renyi (1961). As the reader may e a s i l y check, I (f,g) —> I ( f , g ) as a—>1 whatever be f and g i n A; here, the a r e s t r i c t i o n a < 1 i s imposed to ensure t h a t the i n t e g r a l i n (2.5.4) i s always f i n i t e . Reasoning i n the same way as b e f o r e , we would l i k e to f i n d a p o s s i b l y unique g = P ( f f ) which a n maximizes n Z p I (f ,g), (2.5.5) i=1 i a i the expected Renyi Information of order a, 0 < a < 1. T h i s problem has not yet been s o l v e d f o r a r b i t r a r y n and a. However, a s o l u t i o n f o r the case where n=2 and a = = p 2 =1/2 where (2.5.5) becomes log{ [/v/I7gdti] [ Jt/IIgdii]} i s given below. 75 Lemma 2.5.3 The q u a n t i t y [ jVITgdM] [ jVf"7gclri] H 2/fH 2 du M-a.e., H = /IT + /II. achieves a maximum when g = Proof: Let F, = /IT, F 2 = y/Tl, where H = F, + F 2 and G = /g. We have 2 JF iGd/u SF2Gdu < [;F,Gdy] 2 + [/FjGdiu] 2 and so 4 J"F ! GdM J*F2Gdu ^ [ J(F,+F 2)GdM] 2 * [/(F,+F 2) 2du]-[/G 2dM] = /H 2dy. The second i n e q u a l i t y i s s t r i c t u n l e s s 0 ( H ) 2 = yg u~a.e. f o r some 0 , 7 e R, not both zero (Rudin 1974, p. 66). I f 7 = 0 , then 0 # 0 and so H = 0 j/-a.e., a c o n t r a d i c t i o n . Thus 7 * 0 and g = 0 H 2 / 7 with Jgdj* = 1. Thus /3 * 0 and g = H 2//H 2d*i M~a.e. I t so happens that the f i r s t i n e q u a l i t y i s a l s o achieved by t h i s p a r t i c u l a r c h o i c e of g. • Th i s p a r t i a l r e s u l t was obtained independently by Mr. B.J. Sharpe ( p r i v a t e communication). I t i s he who p o i n t e d out t h a t , i n the s e t t i n g of Example 2.5.5 below, i t i s p o s s i b l e to show n n t h a t , i n g e n e r a l , P * ( I v / T " ' ) 2 / / ( I /T~) 2dM f o r n > 2 and w, a i=1 i i=1 i = ... = w = 1/n. The d e t a i l s are omitted, n We now propose to c h a r a c t e r i z e the f o l l o w i n g p o o l i n g o p e r a t o r s which we c a l l the "normalized (weighted) means of 76 order a:" n a 1 /a n a 1 /a T (f,,...,f ) = [ I w f ] //[ I w f ] du, 0<a<1. (2.5.6) a n i=1 i i i=1 i i (As b e f o r e , the weights w are non-negative and sum up to 1.) i These bear an obvious c o n n e c t i o n with the weighted mean of order a, M ( x x ), of a set of n non-negative r e a l numbers ( c f . a n Hardy, L i t t l e w o o d & P61ya 1934): n a 1 /a M ( x , , . . . , x ) = [ Z w x ] a n i=1 i i These weighted means w i l l appear again i n Chapter 3, when we d i s c u s s the problem of p o o l i n g p r o p e n s i t y f u n c t i o n s . The b a s i c q u a n t i t y , here, i s K u l l b a c k ' s (1968, p. 67) no t i o n of divergence between any two p r o b a b i l i t y d i s t r i b u t i o n s , f and g: a 1-a 6 (f ,g) = 1-Jf g du, 0<a<1. a In the case a - 1/2, 6 (f,g) i s e q u i v a l e n t t o the s o - c a l l e d a H e l l i n g e r (1909) - Kakutani (1948) - M a t u s i t a (1951) d i s t a n c e 77 P 2(f,g) - J(/r - /g) 2d>, a measure also used by Stein (1965) for measuring the distance between posterior d i s t r i b u t i o n s obtained from two d i f f e r e n t prior d i s t r i b u t i o n s . The function p(f,g) i s sometimes referred to as the a f f i n i t y between f and g, after Bhattacharyya (1943). We have the following Theorem 2.5.4 The pooling operator T defined by Equation (2.5.6) i s the only a n one which minimizes the expected divergence I w 5 (f ,g). i=1 i a i Proof: n a Write f = I w f . By Holder's inequality, i=1 i i 1-a 1/a a Jfg dn * [Jf d/u] < » 1/a and equality i s achieved only when /3f = yg u-a.e. for some /3,7 e R, not both zero (Rudin 1974, p. 66). Proceed as in the proof of Lemma 2.5.3. • If a decision maker knows that each one of the n expert opinions f f which he has c o l l e c t e d has a corresponding n p r o b a b i l i t y p of being the "right one," then i t might well seem i 78 reasonable to him to choose a consensus d i s t r i b u t i o n which, on the average, w i l l have the greatest " a f f i n i t y " with the true d i s t r i b u t i o n . In that case, Theorem 2.5.4 above says that T a should be used for some 0 < a < 1. The choice of the value of a may be guided by the s p e c i f i c application; a l t e r n a t e l y , the decision maker might want to assess the s e n s i t i v i t y of his conclusions by computing a consensus for d i f f e r e n t a's. One a t t r a c t i v e feature of the class {T } i s the fact that both the linear and the logarithmic pooling operators are n included as l i m i t i n g cases. Indeed, T ( f f ) -> Z w f as a n i=1 i i a —> 1, whatever f f e A. On the other hand, we can use n L'Hospital's rule to see that n a 1/a -1 n a log[Lim ( Z w f ) ] = Lim a «log[ I w f ] a—> 0 i = 1 i i a—> 0 i = 1 i i n a n a = Lim [ Z w f .log(f )]/[ Z w f ] a—>0 i = 1 i i i i = 1 i i n w ( i ) = log[ n f ]. i = 1 i Now, i t i s known (Hardy, Littlewood & P 6 l y a 1934, p. 26) that n 1/k k n [ Z w f ] £ Z w f i=1 i i i=1 i i 79 n p o i n t w i s e f o r a l l k £ 1, and of course / Z w f du = 1. i = 1 i i T h e r e f o r e , we can use the Lebesgue Dominated Convergence Theorem (Sion 1968, p. 95) to conclude t h a t n 1/k k Lim /[ I w f ] du = k—>°° i=1 i i n 1/k k /Lim [ I w f ] du k—>«> i = 1 i i n w( i ) / n f du. i = 1 i n w(i) n w(i) Consequently, T — > n f / / n f du always as a —> 0. a i=1 i i=1 i T h i s f a c t may pr o v i d e some i n d i c a t i o n t h a t t h i s q u a s i - l o c a l E x t e r n a l l y Bayesian procedure i s "robust" i n some sense. We conclude t h i s s e c t i o n with an example borrowed from Weerahandi & Zidek (1978): Example 2.5.5 (Parliamentary v o t i n g procedures) Suppose that a House of R e p r e s e n t a t i v e s i s composed of n members, each of whom has a democratic weight of 1/n when he vo t e s . Suppose a l s o that when a pr o p o s a l i s put before the House f o r a p p r o v a l , each member i t e l l s an independent judge, Mr. Speaker say, h i s p e r s o n a l p r o b a b i l i t y 0 < p < 1 that i p a s s i n g the pr o p o s a l i s the r i g h t t h i n g t o do. The understanding i s that t h i s person i s r e q u i r e d to form the consensus and take a d e c i s i o n , a p p r o v a l or r e j e c t i o n , which i s 80 c o n s i s t e n t with i t . Note that Mr. Speaker c o u l d have the r i g h t to vote too, as long as he does not l e t h i s p e r s o n a l d e s i r e s i n f l u e n c e unduly the d e c i s i o n u l t i m a t e l y made by him i n h i s c a p a c i t y as a r b i t r a t o r of the group. If Mr. Speaker uses T (0 a < a < 1) to e s t a b l i s h a consensus, h i s a r b i t r a t o r ' s odds i n favour of passing the proposal (once he has heard every deputy) w i l l be n a 1/a n a 1/a [ I p ) /[ Z (1-p ) ] , 0 < a £ 1 i=1 i i=1 i n 1/n n 1/n [ n p ] /[ n (1 -p ) ] , a = 0. i=1 i i=1 i Thus, the best non-randomized d e c i s i o n r u l e would c o n s i s t of pa s s i n g the pro p o s a l i f n a n a (1/n)• Z p > (1/n)• Z (1-p ) , 0 < a S 1; (2.5.7) i=1 i i=1 i n (1/n)- Z l o g [ p /(1-p )] > 0, a = 0. (2.5.8) i=1 i i When a = 1, the procedure reduces t o passing the pro p o s a l i f p > 1/2. When a = 0, the proposal w i l l go through i f , on the average, the p a r l i a m e n t a r i a n s ' l o g - o d d s - r a t i o s favor passage, 81 i . e . (2.5.8) h o l d s . Now suppose that n 0 of the House members, i=1,...,n 0 are a g a i n s t passage, and the other n, = n-n 0 are f o r . F u r t h e r assume that p = 7 < 1/2, i=1,...,n 0, and p = I > 1/2, i=n 0+1, i i ...,n. Weerahandi & Zidek (1978) p o i n t out that i f 7 = 1-£, the optimal non-randomized d e c i s i o n r u l e (2.5.8) i s nothing but the f a m i l i a r "simple m a j o r i t y " v o t i n g procedure. T h i s i s a l s o t r u e of (2.5.7) for a l l 0 < a £ 1, as the reader may e a s i l y check. Thus T 0 i s not e x t r a o r d i n a r y i n t h i s r e s p e c t . But the "simple m a j o r i t y " r u l e would seem w e l l j u s t i f i e d , at l e a s t i f p o l i t i c i a n s were as c e r t a i n as they appear to be i n p u b l i c appearances and t h e r e f o r e the p 's were a l l e s s e n t i a l l y 0 or 1. i 2.6 D i s c u s s i o n The work of the p r e v i o u s s e c t i o n s has been d i r e c t e d toward the t h e o r e t i c a l aspects of group p r o b a b i l i t y assessment i n the case where expert o p i n i o n s are e x p r e s s i b l e as d e n s i t i e s with r e s p e c t to a f i x e d u n d e r l y i n g measure. In p a r t i c u l a r , we have ( i ) proposed new arguments f a v o u r i n g the l i n e a r o p i n i o n p o o l , T,; and ( i i ) c h a r a c t e r i z e d the l o g a r i t h m i c p o o l i n g o p e r a t o r , T 0 , as the "only p r a c t i c a l " ' E x t e r n a l l y Bayesian procedure. We regard the l a t t e r r e s u l t as our main c o n t r i b u t i o n to t h i s problem. Some of the f o l l o w i n g remarks w i l l a l s o apply i n 82 substance to the developments of Chapter 3 (they w i l l not be r e p e a t e d ) . I t should be c l e a r , from the content of S e c t i o n 2.5 e s p e c i a l l y , that there cannot be a unique s o l u t i o n to the problem of the panel of e x p e r t s . T h i s i s a l s o the c o n c l u s i o n reached by Bacharach (1975). C e r t a i n l y , the use of e i t h e r the l i n e a r or the l o g a r i t h m i c pool i s by now w e l l - j u s t i f i e d , and i t i s i n t e r e s t i n g to t h i n k of the two as being l i m i t i n g cases of an e n t i r e c l a s s of reasonable p o o l i n g formulas. T h i s author would p e r s o n a l l y favour the l o g a r i t h m i c p o o l i n g o p e r a t o r , as he f i n d s the axiom of coherence EB of Madansky (1964;1978) r a t h e r a p p e a l i n g i n a Bayesian framework. The p r e s c r i p t i o n n w ( i ) n w ( i ) n f / / n f du (2.6.1) i=1 i i=1 i i s a l s o recommended by Weerahandi & Zidek (1978), and i n d i r e c t l y by M o r r i s (1974;1977) and Winkler (1968), the l a t t e r through h i s n a t u r a l - c o n j u g a t e (N-C) approach. The N-C r e c i p e amounts to that o f f e r e d by the l o g a r i t h m i c o p i n i o n p o o l except that a l l p r o b a b i l i t y assessments must belong to some f i x e d n a t u r a l -conjugate f a m i l y of d i s t r i b u t i o n s . Of course, t h i s approach i s v a l u a b l e only to the extent that such a mathematical model may w e l l approximate one's judgements. Bacharach (1973) a t t r i b u t e s 83 (2.6.1) t o Hammond, but he does not c i t e a source f o r the r e s u l t . As Winkler (1968) p o i n t s out, the c h o i c e of a p o o l i n g operator can be i n f l u e n c e d by p r a c t i c a l c o n s i d e r a t i o n s . For in s t a n c e , the d e s i r e to s i m p l i f y computations or the need to have an a n a l y t i c a l e x p r e s s i o n f o r the consensus may w e l l d e t e r one from using an otherwise s e n s i b l e formula. M o r r i s ' (1977) procedure, f o r example, e n t a i l s formidable assessment problems i n a l l but the sim p l e s t a p p l i c a t i o n s . Thus, the f o l l o w i n g f e a t u r e s of T 0 and T,, the l o g a r i t h m i c and l i n e a r o p i n i o n pools r e s p e c t i v e l y , would be of some releva n c e i n the context of an a c t u a l a p p l i c a t i o n : ( i ) T i i s g e n e r a l l y multi-modal, w h i l s t T 0 i s t y p i c a l l y u n i -modal I t i s g e n e r a l l y observed that the l a r g e r the d i f f e r e n c e s amongst the modes of the i n d i v i d u a l p r o b a b i l i t y d e n s i t i e s f , i the more l i k e l y i t i s that T, w i l l produce a multi-modal d i s t r i b u t i o n . The f a c t t h at T t may f a i l to i d e n t i f y a parameter which t y p i f i e s i t s modes ( i . e . the i n d i v i d u a l c h o i c e s ) might w e l l be p e r c e i v e d as a f a u l t , even i f the problem does not c a l l f o r a d e c i s i o n ( c f . Weerahandi & Zidek 1981). 84 ( i i ) has a greater variance than T 0 This i s not surprising in view of ( i ) . Given a set of w 's, the tighter d i s t r i b u t i o n s w i l l automatically receive more i weight under T 0 than under T,. This i s due to the m u l t i p l i c a t i v e nature of the logarithmic opinion pool. For an analogy, think of the si t u a t i o n faced in a formal Bayesian analysis where a large amount of sample information "swamps" a r e l a t i v e l y smaller amount of pr i o r knowledge. Whether a small or a large variance i s more desirable w i l l depend on the p a r t i c u l a r application one has in mind. Bernardo (1976) reports the following example: suppose that two experts gave f, N(0,1) and f 2 * N(1+/3/e,e2) as the i r respective opinions, and that 8 l a t e r turned out to be 1. Figure 1. Two opinions with a d i f f e r e n t entropy but giving the same p r o b a b i l i t y to the true value of the quantity of interest before i t i s revealed to be 1 . 85 E x p l a i n s Bernardo, on page 3 4 : "In a sample survey where a loose approximation of 6 may be u s e f u l , f 2 c o u l d be p r e f e r r e d on the grounds that i t a t t a c h e s a h i g h p r o b a b i l i t y to such approximate v a l u e s . On the other hand, i n a medical r e s e a r c h where a small e r r o r may have f a t a l consequences, f, c o u l d be p r e f e r r e d on the grounds that i t warns a g a i n s t a premature, p o s s i b l y f a t a l d e c i s i o n and r a t h e r suggests that more evidence i s needed." These p r e o c c u p a t i o n s , however, are somewhat beyond our present concern. ( i i i ) C a l c u l a t i o n s are e a s i e r with T 0 We remark that i f f f are members of a f a m i l y of n e x p o n e n t i a l type determined by the same g e n e r a l i z e d d e n s i t y , then T 0 ( f f ) w i l l be a member of the same f a m i l y . For n example, i f f i s a normal d e n s i t y with mean u and v a r i a n c e a 2 , i i i n then T 0 ( f f ) w i l l a l s o be normal, with mean u = Z a u / n i=1 i i n n Z a and v a r i a n c e o2 = 1/ Z a , where a = w / a 2 , 1<i£n. i=1 i i=1 i i i i n D i s t r i b u t i o n s of the form Z w f are c a l l e d mixtures and i = 1 i i u s u a l l y are i n t r a c t a b l e , u n l e s s of course a l l the f 's are the i same. 86 P r o f e s s o r A.W. M a r s h a l l has r a i s e d an o b j e c t i o n to the l o g a r i t h m i c o p i n i o n pool ( p e r s o n a l communication), p o i n t i n g out that i t would be u n s a t i s f a c t o r y f o r combining expert o p i n i o n s when these o p i n i o n s are based on o v e r l a p p i n g experience or data sources. T h i s i s indeed a problem with Winkler's N-C approach, where (2.6.1) comes as a by-product of Bayes* r u l e . However, t h i s i s not a c r i t i c i s m of the l o g a r i t h m i c o p i n i o n pool which was obtained i n Theorem 2.4.6. I t must be i n t e r p r e t e d i n s t e a d as a c r i t i c i s m of the EB p o s t u l a t e , a l o g i c a l consequence of which i s the l o g a r i t h m i c p o o l . I t may w e l l be that T and the EB p o s t u l a t e ought not to be a p p l i e d to the set {f } f o r the i reasons c i t e d above, but i n s t e a d to an a l t e r n a t e c o l l e c t i o n of o p i n i o n s d e r i v e d from the f 's. However once the d e r i v e d set i s i s p e c i f i e d , EB would again l e a d to t h e i r l o g a r i t h m i c p o o l . M o r r i s (1974,1977) gets around t h i s d i f f i c u l t y by encoding the degree of dependence amongst the experts as w e l l as each expert's p r o b a b i l i t y assessment a b i l i t y i n a s o - c a l l e d J o i n t  C a l i b r a t i o n F u n c t i o n C:© — > [ 0 , ° ° ) . In g e n e r a l , t h i s c a l i b r a t i o n f u n c t i o n w i l l represent the d e c i s i o n maker's s u b j e c t i v e e v a l u a t i o n of the e x p e r t s , r a t h e r than the r e s u l t of the experts e m p i r i c a l l y c a l i b r a t i n g themselves; and so the task of determining C w i l l be rendered d i f f i c u l t by the need t o assess the e l u s i v e dependencies between the e x p e r t s ' o p i n i o n s . 87 When t h i s i s done, a "generalized l o g a r i t h m i c opinion p o o l " M(f ) emerges f o r the composite p r i o r of n experts: n n n M(f,,... ff ) = C n f /JC n f du. (2.6.2) n i=1 i i=1 i This reduces to (2.6.1) with w(i) = 1,1<i<n, i f the experts are independent and c a l i b r a t e d ( i . e . i f C i s a c o n s t a n t ) . This n concurs with the ad hoc suggestion of Winkler (1968) that E w i=1 i should be taken i n the i n t e r v a l [ l , n ] and r e f l e c t the "amount of independence" between the experts. However, M i s not E x t e r n a l l y Bayesian and w i l l u s u a l l y have a much smaller variance than T 0 i f C i s a constant ( c f . remark ( i i ) above). A l s o , note, that M provides us with an example of a "semi-quasi-l o c a l " p o o l i n g operator. I t would be i n t e r e s t i n g to c h a r a c t e r i z e a l l the E x t e r n a l l y Bayesian s e m i - q u a s i - l o c a l p o o l i n g operators. As observed by Winkler (1968) and Weerahandi & Zidek (1981), p o o l i n g operators can be used i n an e n t i r e l y d i f f e r e n t s p i r i t from that which has motivated t h e i r study. Indeed, the assumption that each f represents a s u b j e c t i v e p r o b a b i l i t y i d i s t r i b u t i o n assessed by a member of a panel of experts i s convenient, but not c r u c i a l to the a n a l y s i s . Thus, a s i n g l e i n d i v i d u a l may w e l l choose to r e f l e c t the surmised q u a l i t y of 88 his prior knowledge in an analysis by combining t h i s p r i o r with "mechanical predictions" representing either ignorance or a tendency to persistence (what happened yesterday w i l l happen today), or else derived using more complex schemes such as multiple regression. Nonetheless, the problem of determining appropriate weights remains. F i n a l l y , a word of caution concerning the game-theoretic aspects of our problem. Throughout t h i s chapter, we have assumed that the experts consulted by the decision maker were candid and accurate in their p r o b a b i l i t y assessments. Paraphrasing Rai f f a (1968), we could say that they are dedicated st a f f men, whose s i n c e r i t y i s unquestionable and who would not conspire to t r i c k the decision maker. This i s not a r e a l i s t i c assumption i f the ultimate objective of the exercise i s to make a (possibly consequential) decision and/or i f the decision maker only represents the "synthetic personality" of the group. For example, one of the experts may i n t e n t i o n a l l y f a l s i f y his opinion in an attempt to influence the others toward a p a r t i c u l a r consensus which i s somehow advantageous to himself. Fellner (1965) discusses p r o b a b i l i s t i c "slanting" and i t s occurrence in group decision making. When bargaining i s involved, the solutions presented here w i l l prove unsatisfactory unless, perhaps, the members of the panel share . (roughly) the same preference pattern, i . e . u t i l i t y function. Unlike Weerahandi & Zidek (1981), our approach to the multi-Bayesian decision problem i s through aggregation, not compromise. 89 I I I . POOLING PROPENSITIES 3.1 M o t i v a t i o n Thus f a r , we have concentrated on the problem of r e c o n c i l i n g judgemental p r o b a b i l i t y assessments, i . e . expert opinions which are e x p r e s s i b l e as d e n s i t i e s with respect to some n a t u r a l dominating measure on a space 9 of mutually e x c l u s i v e a l t e r n a t i v e s . In t h i s chapter, we enlarge t h i s problem and concern ourselves with what we c a l l propensity f u n c t i o n s , or P-fu n c t i o n s for sh o r t . Given a space, 0, of contemplated s t a t e s of nature, a P-function i s j u s t a transformation of 0 i n t o (0,»). We w i l l denote by n the set of a l l P-functions on 0. Examples of P-functions are l i k e l i h o o d s , b e l i e f f u n c t i o n s (Shafer 1976) and de n s i t y f u n c t i o n s such as those obtained from p r i o r , vague p r i o r , p o s t e r i o r , s t r u c t u r a l (Fraser 1966) and f i d u c i a l d i s t r i b u t i o n s . These f u n c t i o n s , p, need not have a f i n i t e i n t e g r a l with respect to any p a r t i c u l a r measure; however, they share the property that p(0)/p ( 7 7 ) represents the r e l a t i v e degree of support (or "propensity") expressed i n favour of 8 over 17, 8 and v being elements of 0. This r a t i o p ( t 9)/p ( 7 ? ) may w e l l be an odd s - r a t i o or a l i k e l i h o o d - r a t i o , for example. In any case, the l a r g e r t h i s q u a n t i t y i s , the greater i s the degree of c o n v i c t i o n i n favour of 8 compared to T J . 90 To f i x ideas, some s p e c i f i c examples w i l l now be presented where the need w i l l occur to pool P - f u n c t i o n s . These a p p l i c a t i o n s w i l l a l s o serve as a m o t i v a t i o n f o r the ensuing n developments. In each case, a p o o l i n g o p e r a t o r T:II — > n i s r e q u i r e d . Example 3.1.1 ( p o o l i n g u t i l i t y f u n c t i o n s ) When c o n f r o n t e d with i n t r a - g r o u p c o n f l i c t s , i t i s sometimes necessary to take i n t o c o n s i d e r a t i o n q u e s t i o n s of u t i l i t y . T y p i c a l l y , t h i s w i l l be the case i f a c h o i c e or d e c i s i o n i s to be made which w i l l a f f e c t the members of the p a n e l . I f such a panel of experts d i s a g r e e s on u t i l i t y assignments f o r a c t i o n s (or t h e i r consequences) as w e l l as on the p r i o r p r o b a b i l i t i e s f o r the p o s s i b l e s t a t e s of nature, a d e c i s i o n maker might choose to decompose the problem i n t o two p a r t s , u t i l i t i e s and p r o b a b i l i t i e s , and proceed to e x t r a c t a consensus on both matters independently. T h i s a t t i t u d e i s recommended by R a i f f a (1968, p. 232) d e s p i t e some of i t s shortcomings. Assuming that a s o l u t i o n to t h i s consensus problem i s sought through a g g r e g a t i o n , the d e c i s i o n maker w i l l have to decide on a formula f o r amalgamating the experts' u t i l i t y f u n c t i o n s . In that r e s p e c t , the techniques of Chapter 2 are of no a v a i l because u t i l i t y f u n c t i o n s g e n e r a l l y do not i n t e g r a t e to one. On the other hand, note t h a t any s t r i c t l y p o s i t i v e u t i l i t y f u n c t i o n i s a P - f u n c t i o n , so that - i n that case at l e a s t -91 p o o l i n g u t i l i t y f u n c t i o n s amounts to p o o l i n g P - f u n c t i o n s . Furthermore, the p o s i t i v i t y assumption can be waived i f we adopt the P a r e t i a n a t t i t u d e that a measure of u t i l i t y i s o r d i n a l (as opposed to c a r d i n a l ) , and hence unique only up to a s t r i c t l y monotonic i n c r e a s i n g t r a n s f o r m a t i o n (of course, not a l l such u t i l i t y f u n c t i o n s w i l l s a t i s f y the e x p e c t e d - u t i l i t y p r i n c i p l e ) . Example 3.1.2 ( p o o l i n g l i k e l i h o o d s ) In g e n e r a l , l i k e l i h o o d f u n c t i o n s are d e r i v e d from c o n d i t i o n a l p r o b a b i l i t y d i s t r i b u t i o n s of the form P{data|true value i s 6} c o n s i d e r e d as a f u n c t i o n of 8. These c o n d i t i o n a l d i s t r i b u t i o n s can be s u b j e c t i v e . F or, i n a l l but the s i m p l e s t s t a t i s t i c a l a p p l i c a t i o n s , they are modelled i n some convenient way to approximate more or l e s s a c c u r a t e l y the observed (but unknown) u n d e r l y i n g d i s t r i b u t i o n of the data. T h i s mathematical mo d e l l i n g i n v o l v e s some i n t r o s p e c t i o n and a r b i t r a r i n e s s on the p a r t of the a s s e s s o r , and s u b j e c t i v e l i k e l i h o o d s are a r e f l e c t i o n of th a t m i l d " i n t e r p r e t a t i o n " of f a c t s and evidence. If a problem were d e l i c a t e and complicated enough, i t would not be s u r p r i s i n g that a panel of experts who were not i n accor d on p r i o r p r o b a b i l i t i e s f o r the p o s s i b l e s t a t e s of nature were not i n agreement e i t h e r on the meaning and/or value of some new p i e c e of i n f o r m a t i o n presented to them. At best, i n those cases where the data was r e l i a b l e , of good q u a l i t y and r e l a t e d to the 92 parameter of i n t e r e s t i n a manner which i s w e l l understood, even c r i t i c a l e x p e r t s would probably n e a r l y agree on t h e i r s i g n i f i c a n c e and might even adopt the a s s o c i a t e d l i k e l i h o o d as t h e i r (common) " r e v i s e d o p i n i o n . " However, i t i s easy to conce i v e of s i t u a t i o n s where the new i n f o r m a t i o n would be so much s u b j e c t to p e r s o n a l i n t e r p r e t a t i o n that i t s d i s c l o s u r e would cause the experts to d i s a g r e e even f u r t h e r ! T h i s suggests circumstances i n which expert p r o b a b i l i t y a s s e s s o r s c o u l d be l e f t i n d i s s e n s u s , even a f t e r an open and vigorous exchange of informat i o n . One method f o r r e s o l v i n g the disagreement amongst experts about the i n t e r p r e t a t i o n of a set of data i s to pool t h e i r a s s o c i a t e d s u b j e c t i v e l i k e l i h o o d f u n c t i o n s . When these l i k e l i h o o d s are non-zero everywhere on 0 (the usual case and the most i n t e r e s t i n g one), t h i s reduces to the problem of agg r e g a t i n g p r o p e n s i t i e s . Example 3.1.3 1 (combining independent t e s t s of hypot h e s i s ) T h i s i s a well-known problem which has been i n v e s t i g a t e d by many authors; f o r a g e n e r a l d i s c u s s i o n and a f a i r l y e x t e n s i v e b i b l i o g r a p h y on the s u b j e c t , we r e f e r to Monti & Sen (1976). The formal s t a t i s t i c a l problem may be s t a t e d as f o l l o w s : given n 1 We are t h a n k f u l to Dr. Peter McCullagh of Imp e r i a l C o l l e g e (London) f o r suggesting t h i s a p p l i c a t i o n . 93 independent t e s t s t a t i s t i c s , J 1 f . . . , J , f o r t e s t i n g a n u l l n h y p o t h e s i s H0:weS20 versus H^rcjefl, (0 = ft0U&i being a space of p r o b a b i l i t y d i s t r i b u t i o n s ) , s e l e c t a f u n c t i o n , P, of J 1 f . . . , J n which i s to be used as the combined t e s t s t a t i s t i c . The idea behind the c o n s t r u c t i o n of P i s that the aggregate of s e v e r a l t e s t s , p o s s i b l y of marginal s i g n i f i c a n c e i n d i v i d u a l l y , can l e a d to s c i e n t i f i c a l l y d e c i s i v e c o n c l u s i o n s i f t h e i r r e s u l t s are viewed as a whole. I f l a r g e v a l u e s of the J 's are c o n s i d e r e d i c r i t i c a l f o r t e s t i n g H 0, one common s o l u t i o n c o n s i s t s of f i n d i n g a t e s t P based on the observed s i g n i f i c a n c e l e v e l s or P-values, L - 1-F (J ), where F (t) = P 0 ( J <t}, the cumulative i i i i i d i s t r i b u t i o n of J under the n u l l hypothesis ( i t i s assumed that i the p r o b a b i l i t y d i s t r i b u t i o n of each J i s the same f o r a l l w e i fi0)• For example, F i s h e r ' s (1932) omnibus procedure i s given by n 1/n P ( L 1 f . . . , L ) = n L and H 0 i s r e j e c t e d when the observed n i = 1 i value of P i s s m a l l . DeGroot (1973) has shown that i t i s p o s s i b l e to i n t e r p r e t the t a i l area L as a p o s t e r i o r p r o b a b i l i t y or as a l i k e l i h o o d i r a t i o f o r the acceptance of the n u l l . Because of t h i s , each L i may be regarded as an i n d i v i d u a l e x p r e s s i o n of b e l i e f or P-f u n c t i o n a s s e s s i n g the " l i k e l i h o o d " of 0 = {"H0 i s true"} (the 94 u n i n t e r e s t i n g case where L = 0 being n e g l e c t e d ) . In that case, i n n P ( L 1 f . . . , L ) i s a p o o l i n g operator a c t i n g on A = (0,1) . n Example 3.1.4 (The Bergson-Samuelson s o c i a l - w e l f a r e f u n c t i o n ) Economists make a d i s t i n c t i o n between what they c a l l an i n d i v i d u a l ' s s o c i a l - w e l f a r e f u n c t i o n and h i s u t i l i t y f u n c t i o n ( c f . Samuelson 1947, Chapter 8 ) . A person's u t i l i t y f u n c t i o n q u a n t i f i e s , e i t h e r c a r d i n a l l y or o r d i n a l l y , what they p r e f e r on the b a s i s of t h e i r p e r s o n a l i n t e r e s t s or on any other b a s i s . On the other hand, the s o c i a l - w e l f a r e f u n c t i o n i s supposed to express what the i n d i v i d u a l p r e f e r s (or, r a t h e r , would p r e f e r ) on the b a s i s of impersonal s o c i a l c o n s i d e r a t i o n s a l o n e . Stated d i f f e r e n t l y , the s o c i a l - w e l f a r e f u n c t i o n r e p r e s e n t s the i n d i v i d u a l ' s " e t h i c a l " or "moral" p r e f e r e n c e s , whereas the u t i l i t y f u n c t i o n d e s c r i b e s t h e i r " s u b j e c t i v e " p r e f e r e n c e p a t t e r n . I t i s the former which the person would use i f they were c a l l e d upon to make a moral value judgement. Mathe m a t i c a l l y , a s o c i a l - w e l f a r e f u n c t i o n i s a mapping W which makes correspond a " s o c i a l u t i l i t y " W(u,,...,u ) to any n v e c t o r (u,,...,u ) of p r i v a t e u t i l i t i e s . P rovided that i t s n domain and range are a p p r o p r i a t e l y r e s t r i c t e d , W i s an in s t a n c e of a P - f u n c t i o n p o o l i n g o p e r a t o r . The q u e s t i o n of d e f i n i n g and determining the form of a "reasonable" s o c i a l - w e l f a r e f u n c t i o n 95 has occupied economists f o r some time ( c f . , e . g . , Harsanyi 1955). G e n e r a l l y speaking, there i s agreement on two p o i n t s : ( i ) W should be " l o c a l , " i . e . the s o c i a l u t i l i t y l e v e l a t t a c h e d to a "prospect" should only depend on the i n d i v i d u a l u t i l i t i e s a s s o c i a t e d with that p a r t i c u l a r p r o s p e c t ; and ( i i ) W should be i n c r e a s i n g i n each of i t s arguments, the r a t i o n a l e being that " i f you i n c r e a s e any agent's u t i l i t y without d e c r e a s i n g anybody e l s e ' s u t i l i t y , then s o c i e t y i s made b e t t e r o f f . " As we s h a l l see, these two c o n d i t i o n s p l a y an important r o l e i n the seq u e l . Def i n i t i o n 3.1.5 n A p o o l i n g operator T:I1 —>IT i s c a l l e d l o c a l whenever there n e x i s t s a f u n c t i o n G:(0,°°) — > ( 0 , « ) such t h a t T(p,,...,p )(6) = G(p,(6),...,p (0)) (3.1.1) n n for a l l 6e& and p 1 f . . . , p e II. n Note that Equation (3.1.1) must h o l d everywhere, and not merely "almost everywhere." T h i s i s rendered necessary by the absence of any n a t u r a l c h o i c e f o r a dominating measure on ©. The f o l l o w i n g lemma g i v e s obvious e q u i v a l e n t c o n d i t i o n s f o r an operator to be l o c a l . 96 Lemma 3.1,6 n A p o o l i n g operator T:II — > II i s l o c a l i f f ( i ) T ( p 1 f — ,p )(0) = T(p,,...,p )(TJ) whenever p (0) = p (TJ) f o r n n i i a l l i=1,...,n; and ( i i ) T(p,,...,p )(0) = T(q,,...,q )(0) whenever p (0) = q (0) n n i i f o r a l l i=1,...,n. Proof: T h i s i s t r i v i a l . • C o n d i t i o n ( i ) above c o u l d be c a l l e d " c o n s i s t e n c y . " I f a l l the e x p e r t s agree t h a t s t a t e s 0 and TJ are "equiprobable" or " e q u a l l y l i k e l y , " then, a c c o r d i n g t o C o n d i t i o n ( i ) , the d e c i s i o n maker should a t t r i b u t e the same " l i k e l i h o o d " or " p r o p e n s i t y " to both 0 and TJ. To assume that the d e c i s i o n maker i s c o n s i s t e n t in t h i s p a r t i c u l a r sense seems n o n - c o n t r o v e r s i a l , at l e a s t i f i t i s b e l i e v e d that the a s s e s s o r s d i d not b i a s t h e i r judgements i n the hope of g a i n i n g some s t r a t e g i c advantage. C o n d i t i o n ( i i ) can be regarded as a l i k e l i h o o d p r i n c i p l e f o r P - f u n c t i o n s , j u s t as before ( c f . page 29). D e f i n i t i o n 3.1.7 n A p o o l i n g operator T:I1 — > II i s s a i d to preserve the o r d e r i n g of  b e l i e f s (POB) i f f ( i ) T ( p 1 f . . . , p )(0) <. T ( p 1 f . . . , p )( T?) whenever p (0) < p (T J) f o r n n i i 97 a l l i=1 ,...,n; and ( i i ) T(p,,...,p )(0) < T(p,,...,p )(TJ) whenever p (0) < p (77) n n i i f o r a l l i=1,...,n with s t r i c t i n e q u a l i t y f o r some i . The above p r o p e r t y corresponds to the second requirement set out by economists f o r the s o c i a l - w e l f a r e f u n c t i o n . Apart from i t s c o n s i d e r a b l e i n t u i t i v e appeal, we w i l l f i n d that i t o f t e n a c t s as a " r e g u l a r i t y c o n d i t i o n " i n a manner s i m i l a r to the m e asurabi1ity assumption used i n Chapter 2. Note that any POB p o o l i n g operator w i l l s a t i s f y C o n d i t i o n ( i ) of Lemma 3.1.6. In S e c t i o n 3.2 below, l o c a l i t y w i l l be used i n c o n j u n c t i o n with four p o s t u l a t e s of r a t i o n a l i t y i n order to c h a r a c t e r i z e a l a r g e c l a s s of P - f u n c t i o n p o o l i n g o p e r a t o r s . Amongst the requirements w i l l appear the Unanimity P r i n c i p l e , which says that p, = ... = p = p => T(p,,...,p ) = p n n whatever p e n (see Axiom A below). T h i s c o n d i t i o n only makes sense i f the s c a l e s of b e l i e f used by the d i f f e r e n t e x p e r t s are intercomparable, i . e . i f there e x i s t s an " o u t s i d e " standard f o r the q u a n t i f i c a t i o n of b e l i e f such as r e q u i r i n g t h a t the most p r e f e r r e d a l t e r n a t i v e be a s c r i b e d a value of one. The d i f f i c u l t i e s a s s o c i a t e d with comparing degrees of b e l i e f are mentioned i n Weerahandi & Zidek (1981); they are analogous to 98 those which aris e in the theory of u t i l i t y when one attempts to compare "preferences." Although various approaches have been taken to overcome th i s problem in the l a t t e r context, the question remains largely unsolved (cf. Luce & Rai f f a 1957; Sen 1970). In the case of degrees of b e l i e f , the d i f f i c u l t i e s which are referred to above derive from the existence of p o s s i b i l i t i e s which have not yet been i d e n t i f i e d and which, therefore, are not included in 0. Situations are conceivable where i t would be natural to normalize p as p /p (60), B0 being some fixed and i i i distinguished state in 0. In others, p /Sup{p (6)\6eQ} might be i i more natural. In others s t i l l , there might be a natural dominating measure u on 0 with respect to which every p could i be normalized; t h i s , of course, i s the very important special case which we discussed in Chapter 2. In general though, no p a r t i c u l a r choice seems dictated. Furthermore, certain a l t e r n a t i v e s would not always be feasible as, for example, the above-mentioned d i v i s i o n by p 's t o t a l mass when 0 has an i i n f i n i t e M~measure. Notwithstanding these problems associated with the intercomparability of scales of b e l i e f , we s h a l l assume for the time being that i t i s reasonable to use (local) unanimity preserving pooling operators for combining P-functions. In Section 3.3, an attempt w i l l be made at solving the more d i f f i c u l t problem involving the pooling of incomparable 99 p r o p e n s i t i e s . 3.2 A c l a s s of l o c a l p o o l i n g o p e r a t o r s We now propose c e r t a i n weak and ap p e a l i n g c o n d i t i o n s which, a p a r t from l o c a l i t y , embrace the minimal requirements that any reasonable candidate f o r the r o l e of p o o l i n g operator should s a t i s f y (from now on, i t i s understood that a p o o l i n g operator a c t s on P - f u n c t i o n s ) . These "axioms" are seen to c h a r a c t e r i z e the q u a s i - a r i t h m e t i c weighted means ( d e f i n e d below), a r e s u l t which was proven i n another context by A c z e l (1948). An i n t e r e s t i n g f e a t u r e of the theorem i s that although a l l q u a s i -a r i t h m e t i c means are continuous i n t h e i r n v a r i a b l e s , no assumption of smoothness appears i n the l i s t of axioms. When c o n s i d e r a t i o n s r e l a t i n g to the s c a l e s of b e l i e f are added, a c h a r a c t e r i z a t i o n of the l i n e a r and the l o g a r i t h m i c p o o l i n g o p e r a t o r s are obt a i n e d . D e f i n i t i o n 3.2.1 n A t r a n s f o r m a t i o n T:I1 —>IT i s c a l l e d a q u a s i - a r i t h m e t i c p o o l i n g  operator i f f there e x i s t s a continuous and s t r i c t l y i n c r e a s i n g f u n c t i o n i//:(0,°°)—> R with i n v e r s e i K 1 such that n T(p,,...,p ) = </r1[ Z w <Mp )1 (3.2.1) n i=1 i i n f o r some f i x e d weights £ 0 with Z w = 1. n i= 1 i 100 Important examples of q u a s i - a r i t h m e t i c p o o l i n g operators n are the l i n e a r opinion pool Z w p [\p{x)=x], the l o g a r i t h m i c i=1 i i n w ( i ) opinion pool n p [i//(x)=log(x) ] and the root-mean-power i=1 i n c 1/c c poo l i n g operator ( Z w p ) [v//(x)=x ,c>0] which includes the i = 1 i i f i r s t one as a s p e c i a l case (C=1) and the second as a l i m i t i n g case (c->0). The basic p r o p e r t i e s of the qu a n t i t y (3.2.1) are discussed i n Hardy, L i t t l e w o o d & P 6 l y a (1934) i n the case where the p 's are r e a l numbers. E s p e c i a l l y noteworthy i s the f a c t i t hat the f u n c t i o n v// i s unique only up to an order-preserving a f f i n e transformation ax+b, a > 0. This r e s u l t we record as Lemma 3.2.2 n Let w,,...,w > 0 be f i x e d with Z w =1 and, for j=1,2, l e t n i = 1 i n G (x) = }p~ 1 [ Z w \p (x ) ] be two q u a s i - a r i t h m e t i c weighted means j j i = 1 i j i such that G 1(x)=G 2(x) whenever x e l for a l l i=1,...,n, I i being some open i n t e r v a l i n (0,°°). I f there e x i s t at l e a s t two n s t r i c t l y p o s i t i v e w 's, then i / / 2 = a i^+b on I for some a,b e R , i a>0. 101 Proof; T h i s r e s u l t i s s t a t e d and proved as Theorem 83 on page 66 i n Hardy, L i t t l e w o o d & P61ya (1934). • We w i l l now present four axioms which p o o l i n g operator c o u l d reasonably be r e q u i r e d to s a t i s f y . I t w i l l turn out that these axioms c h a r a c t e r i z e the q u a s i - a r i t h m e t i c p o o l i n g o p e r a t o r s of D e f i n i t i o n 3.2.1. The c e n t r a l requirement i s i n s p i r e d by Weerahandi & Zidek's (1978) " p r i o r - t o - p r i o r coherence" axiom and s t i p u l a t e s that " p o o l i n g o p i n i o n s can be done s e q u e n t i a l l y and i n any o r d e r . " Since the u l t i m a t e o b j e c t i v e of p o o l i n g P - f u n c t i o n s i s to c o n s t r u c t something that can be c a l l e d a combined P - f u n c t i o n r e p r e s e n t i n g the b e l i e f s of a l l the e x p e r t s , i t i s p l a u s i b l e that the a c t u a l order i n which t h i s p o o l i n g i s done should be i m m a t e r i a l . In t h e i r manuscript, Weerahandi 6 Zidek express t h i s c o n d i t i o n as T ( p 1 f . . . , p ) = T 2 ( T (p,,...,p ),p ) (3.2.2) k k k-1 k-1 k f o r a l l k=2,...,n, where the s u b s c r i p t on T i n d i c a t e s the dimension of i t s domain. Although i t conveys the b a s i c idea of s e q u e n t i a l p o o l i n g , t h i s f o r m u l a t i o n of the p r i o r - t o - p r i o r coherence axiom i s inadequate because i t may induce an u n d e s i r e d f u n c t i o n a l r e l a t i o n s h i p amongst the weights a s c r i b e d to the v a r i o u s e x p e r t s . T h i s p o i n t w i l l be best i l l u s t r a t e d with an example. 102 Suppose that we are d e a l i n g with n=3 experts and tha t expert i ' s o p i n i o n , p , has a weight of w >0, w,+w2+w3=1. Then i i the r e l a t i v e weight of op i n i o n p, with respect to p 2 i s w = w 1 / < w 1 + w 2 ) r so T 2 ( p 1 , p 2 ) must a s s i g n a weight of w to i t s f i r s t component. However, expert 3 has a weight of w3, which i m p l i e s that the combined o p i n i o n of exp e r t s 1 and 2, p, should c a r r y a weight of w,+w2. Since T 3 ( p , , p 2 , p 3 ) = T 2 ( p , p 3 ) , i t f o l l o w s that w,+w2 = w = w 1/(w,+w 2), from which we conclude that w2 = j / w 7 _ w , and i n turn w3 = w, being a r b i t r a r y i n (0,1). In other words, the weight a t t r i b u t e d to expert 1 determines the weights of both e x p e r t s 2 and 3, and not only t h e i r sum!! T h i s i s c l e a r l y unacceptable. In the above example, the d i f f i c u l t y arose from an ambiguity i n the use of T 2 as i t appears i n Equation (3.2.2) when k=3. There, the "inner T 2 " has the o p i n i o n of expert i as i t s i - t h component,i=1,2, whereas the j o i n t o p i n i o n of exp e r t s 1 and 2 appears i n the f i r s t s l o t of the "outer T 2." To av o i d t h i s problem, i t w i l l be necessary to d i s t i n g u i s h a l l p o o l i n g o p e r a t o r s by keeping t r a c k of the s p e c i f i c weights which they a s c r i b e to t h e i r v a r i o u s components, the u n d e r l y i n g idea being that the source of an o p i n i o n i s i r r e l e v a n t as long as i t has been p r o p e r l y c a l i b r a t e d . 103 Change i n n o t a t i o n When a p p r o p r i a t e , we s h a l l w r i t e T ( p i , . . . , P I ) to n n n denote the j o i n t o p i n i o n of n exp e r t s E,,...,E whose p r o p e n s i t y n f u n c t i o n s , p , are to be weighted and amalgamated using formula i n T and c e r t a i n weights w £0, i=1,...,n, Z w = 1 . n i i=1 i If T i s l o c a l , we s h a l l w r i t e T (p,,...,p | )(d) = n n n n G ( p ^ G ) , . . . , ? (0)|w,,...,w ). n n n In adopting t h i s c onvention, i t i s understood t h a t T n should s a t i s f y T (p,,...,p |w,,...,w ) = n n n T (p ,...,p |w ,...,w ) (3.2.3) n T(1) T(1) T(1) T(1) where T i s any permutation of the set {l,...,n}. S t r i c t l y speaking, t h i s requirement c o u l d be i n c l u d e d i n the f o l l o w i n g l i s t of axioms. However, we p r e f e r to regard i t as an i n t r i n s i c p r o p e r t y of a l l P - f u n c t i o n p o o l i n g o p e r a t o r s . Axiom A (Unanimity P r i n c i p l e ) T (p,, ,p |w,,...,w )= p whenever p, = ... = p = p n n n n 104 Axiom B ( P r e s e r v a t i o n of the o r d e r i n g of b e l i e f s ) T ( p 1 f . . . , p | w w )(6) < T ( q , , — , q |w 1 f—,w )(0) n n n n n n whenever p (d) £ q (0) f o r a l l 1<i£n, the i n e q u a l i t y being i i s t r i c t i f , i n a d d i t i o n , p (d) < q (0) f o r some 1^j^n with j j 0<w <1. When T i s l o c a l , t h i s i s e q u i v a l e n t to saying that T j n n i s a POB p o o l i n g o p e r a t o r . Axiom C ( P r i o r - t o - p r i o r coherence) For a l l 1<k<n, T (p,, ,p |w,, ,w ) = T (p,p ,...,p |w,w ,...,w ) n n n n-k+1 k+1 n k+1 n k where p = T ( p 1 f . . . , p |w,/w,...,w /w) and w = I w ( i f w=0, p k k k i=1 i i s an a r b i t r a r y P - f u n c t i o n ) . Axiom D (Monotonicity of weights) If w <w*, w = 1-w +w* and there e x i s t s j * i such that w >0, then i i i i j T ( p p | w, , — , w )(0) < n n n T ( p p |w,/w,...,w*/w,—,w /w)(6) n n i n provi d e d t h a t p (6) = max{p (0)|l<k<n & w >0}. i k k 105 When n>4, we have the f o l l o w i n g Theorem 3.2.3 ( A c z e l 1948) Let T be l o c a l and suppose that there e x i s t T 2,...,T such n n-1 that Axioms A-D above be s a t i s f i e d . Then T i s a q u a s i -n a r i t h m e t i c p o o l i n g o p e r a t o r . Remark 3.2.4: The essence of the proof i s c o n t a i n e d i n A c z e l ' s (1948) paper. However, we have adapted h i s axioms and f r e e d them from some obvious redundancies. For completeness, the necessary a d a p t a t i o n of h i s proof i s given below. Proof of Theorem 3.2.3: F i r s t , we show that T 2,...,T are l o c a l whenever T i s . For n-1 n k t h a t , we d e f i n e n-2 f u n c t i o n s G :(0,») — > ( 0 , » ) , k=2,...,n-1 by k l e t t i n g G ( y i , . . . , y |v,,...,v ) = k k k T (p,,...,p ,...,p |v,,...,v ,O,...,O)(0) n k k k f o r any p 1 f . . . , p ell with p (0)=y ,1<i^n. Since T i s l o c a l , the k i i n G 's are w e l l - d e f i n e d . Furthermore, we can use p r i o r - t o - p r i o r k 106 coherence to see that T (pi,...,p ,...,p |v,,...,v ,0,...,0)(0) = n k k k T (p,,...,p ,p|v,,...,v ,v)(0) k k-1 k-1 where p = T (p ,...,p |v ,0,...,0) = p by unanimity and n-k+1 k k k k v = v + 0 + . . . + 0 = v . Therefore, k k T (p,,...,p |v,,...,v )(0) = G (p,(0),...,p (0)|v 1 f...,v ) k k k k k k always, i . e . T i s l o c a l for a l l k=2,...,n. k Next, we define a function X:[0 , 1 ]—>[a,b] and v e r i f y that G 2(x(s) ,x(t) | 1-w,w) = x[d-w)s+wt] (3.2.4) for a l l s,t,w € [0,1 ]. In fact, as i s shown below, we may l e t x(«) = G2(a,b|1-w,w) for a l l 0<W<1, so that x(0)=a and x(1)=b by unanimity and p r i o r - t o - p r i o r coherence. Using Axioms A and B, we see that a = G2(a,a|1-w,w) < x(w) < G2(b,b|1-w,w) = b for a l l w e ( 0 , 1 ) , and i t follows from Axiom D that x i s s t r i c t l y increasing in ( 0 , 1 ) . Conjugating Equation (3.2.3) with Axioms A and C, we have successively G 2(x(s),x(t)|1-w,w) = G 2[G 2(a,b|1-s,s),G 2(a,b|1-t,t)|1-w,w] = G 3[a,b,G 2(a,b|1-t,t)|(1-w)(1-s),(1-w)s,w3 = Gja,b,a,b|(l-w)(l-s),( l-w.^s, (l-t)w,tw] = G«[a,a,b,b|(1-w)(1-s),(1-t)w,(1-w)s,tw) = G 3[a,b,b|1-s+ws-wt,(1-w)s,tw3 = G2[a,b|1-s+ws-wt,s-ws+wt] 107 = x(s-ws+wt) = xf(1-w)s+wt], so t h a t (3.2.4) holds t r u e . The key o b s e r v a t i o n i s t h a t x i s continuous on [0 , 1 ] . For then i t i s s u r j e c t i v e on [a,b] by the Intermediate Value Theorem and hence i t has an i n v e r s e x _ 1 : [ a , b ] — > [ 0 , 1 ] . T h i s allows us to r e w r i t e Equation (3.2.4) as G 2(y,z|1-w,w) = x [ ( 1 - w ) x " 1 ( y ) + w X ~ 1 ( z ) ] (3.2.5) where y and z are i n [a,b] and 0^w<1. (x~ 1 w i l l be the f u n c t i o n \p of Equation (3.2.1).) We argue f o r x's c o n t i n u i t y by c o n t r a d i c t i o n . Suppose t 0 i s a p o i n t of d i s c o n t i n u i t y of x, say to the r i g h t . Then 3 ye[a,b] V s > 0 ( s + t o e [ 0 , 1 ] = > x ( t 0 ) < y < x ( t 0 + s ) ) . For such a number y e [ a , b ] , w r i t e y = G 2[X(t),y|1/2,1/2] f o r t a l l t e [ 0 , l ] . By Axiom B, we have G 2 [ x ( t ) , x ( t 0 ) | 1 / 2 , 1 / 2 ] < G 2[x(t),y|1/2,1/2] = y < G 2 [ X ( t ) , x ( t 0 + s ) | 1 / 2 , 1 / 2 ] , t i . e . , using Equation (3.2.4), x [ ( t + t 0 ) / 2 ] < y < x t ( t + t 0 + s ) / 2 ] t f o r a l l s>0 with (t+t 0+s)/2 e [0 , 1 ] . T h i s would show that x i s d i s c o n t i n u o u s at a l l ( t + t 0 ) / 2 e [0,1]; however, a monotone f u n c t i o n never has more than a countable number of d i s c o n t i n u i t i e s (Theorem 4.30; Rudin 1976, p. 96). Hence, x i s continuous everywhere and Equation (3.2.5) o b t a i n s . U s i n g i n d u c t i o n , we w i l l now prove that k G ( y i , . l . , y | v 1 f . . . , v ) = x [ Z v x " 1 ( y )] k k k i=1 i i 108 k for a l l 2^k<n and y c [a,b], v £0, Z v = 1. Indeed, we i i i=1 i deduce from Axiom C that G (yi,.««,Y |v 1 f..,v ) equals k+1 k+1 k+1 k y i f v = I v = 0 and G 2[G (yi,...,Y Ivt/v,...^ /v),y |v, k+1 i = 1 i k k k k+1 v ] otherwise. k+1 However, we know by hypothesis that k G (yi,...,y |v,/v,...,v /v) = xt Z (v /v)-x" 1(y )] k k k i = 1 i i . and so G (yi,...,y | v i,...,v ) equals k+1 k+1 k+1 k G 2[x{ Z (v /v).x* 1(y )},y |v,v ]. i=1 i i k+1 k+1 Using Equation (3.2.5) now, we fi n d that t h i s l a s t quantity equals k x[v. X" 1{x( Z (v / v ) . x _ 1 ( y ))} + v - x _ 1 ( y ) ] , i=1 i i k+1 k+1 k+1 which i s nothing but xt Z v x" 1(y ) ] . i = 1 i i To complete the proof, i t remains to show that i f yi,...,y are n any in (0,»), then ^:(0,»)—>R exists which i s continuous, s t r i c t l y increasing, and such that n G (yi,...,y |w1f...,w ) = tf-'l Z w t//(y )] (3.2.6) n n n i=1 i i n with w,,...,w ^ 0, Z w = 1 . n i = 1 i 109 For t h i s , we c o n s i d e r the nested sequence of c l o s e d i n t e r v a l s I m = [1/m+l,m+1] i n (0,°°); we can repeat the argument above with a=l/m+1 and b=m+1 to prove the e x i s t e n c e of a continuous and s t r i c t l y i n c r e a s i n g f u n c t i o n x" 1 = ^ :I — > [ 0 , 1 ] such that m m m Equation (3.2.6) holds f o r y i , . . . , y e I and $ i n s t e a d of \//, n m m n m=1,2... Since I, i s con t a i n e d in I and 1 [ Z w V/,(y )] = m 1 i=1 i i n \p~1 [ Z w \jj (y ) ] on I , , we conclude from Lemma 3.2.2 that ii = m i=1 i m i m a \//, + b f o r some a ,b e R with a s t r i c t l y p o s i t i v e . D efine m m m m m yfr* = - b ]/a so th a t ^* i s an extension of ^ ^ on I , m>2; we m m m m m m have that ^* extends from I to I , s i n c e \p* = a^* + b m+1 m m m+1 m+1 m fo r some a,b e R with a>0, and \p* = i//* = i / ^ on I , ( i . e . m+1 m a=1,b=0). We l e t \p = ^* on I , m=1,2... m m Th i s completes the proof of Theorem 3.2.3. • Theorem 3.2.3 p r o v i d e s s t r o n g t h e o r e t i c a l support f o r using a q u a s i - a r i t h m e t i c p o o l . If a d e c i s i o n maker wants h i s p o o l i n g operator to be l o c a l , to preserve unanimity and the o r d e r i n g of b e l i e f s , as w e l l as to s a t i s f y the two eminently reasonable axioms of p r i o r - t o - p r i o r coherence and monotonicity of weights, then he must use Formula (3.2.1), with some undetermined i / / , to pool h i s ex p e r t s ' P - f u n c t i o n s . The requirement that n£4 i s not 110 r e a l l y l i m i t i n g , because one can always throw i n dummy o p i n i o n s and a s s i g n them a zero weight. The only p o s s i b l y c o n t r o v e r s i a l h y p o t h e s i s , t h e r e f o r e , i s that the v a r i o u s s c a l e s of b e l i e f are intercomparable; t h i s i s e s s e n t i a l i n order f o r the Unanimity P r i n c i p l e t o make sense ( c f . S e c t i o n 3.1). As long as t h i s assumption i s v a l i d , i t seems f a i r t o say that A c z e l ' s c o n d i t i o n s r e f l e c t the minimal requirements t h a t any s e r i o u s candidate to the r o l e of P - f u n c t i o n p o o l i n g operator would be expected to meet. How should one choose the f u n c t i o n $ i n Formula (3.2.1)? C l e a r l y , there i s no unique way to answer t h i s q u e s t i o n , at l e a s t not at t h i s l e v e l of a b s t r a c t i o n . G e n e r a l l y speaking, the c r i t e r i o n f o r s e l e c t i n g a "good" <// w i l l vary depending on the a p p l i c a t i o n at hand together with the intended g o a l . In c e r t a i n circumstances, a l l c h o i c e s of \(/ w i l l be m e r i t o r i o u s ; t h i s w i l l happen i n the problem of p o o l i n g P-values ( c f . Theorem 3.2.9 below). In other cases, however, s p e c i a l c o n s i d e r a t i o n s r e l a t i n g to the s c a l e s of b e l i e f may induce some symmetry or i n v a r i a n c e i n the problem, and -as a r e s u l t - i t c o u l d seem reasonable t o r e s t r i c t the c l a s s of q u a s i - a r i t h m e t i c p o o l i n g o p e r a t o r s t o those o p e r a t o r s which are symmetric or i n v a r i a n t with r e s p e c t to c e r t a i n o p e r a t i o n s . T h i s method i s f r e q u e n t l y used i n s t a t i s t i c a l d e c i s i o n theory f o r choosing a d e c i s i o n r u l e in cases where an o v e r a l l best r u l e does not e x i s t . As we s h a l l see p r e s e n t l y , i t w i l l prove s u c c e s s f u l when s e a r c h i n g f o r a Bergson-Samuelson s o c i a l - w e l f a r e f u n c t i o n , amongst o t h e r s . 111 Examples 3.1.1 & 3.1.4 (continued) In t h i s example, we are concerned with f i n d i n g an ac c e p t a b l e j o i n t u t i l i t y f u n c t i o n , W(u 1 f...,u ), which might n w e l l be a Bergson-Samuelson s o c i a l - w e l f a r e f u n c t i o n . For that purpose, we make the somewhat r e s t r i c t i v e assumptions that the u 's are c a r d i n a l u t i l i t y f u n c t i o n s which are bounded from below i and, more i m p o r t a n t l y , which are intercomparable. The f i r s t c o n d i t i o n i m p l i e s that u +c > 0 f o r some ceR independent of i i and so the u 's can be simply t r e a t e d as p r o p e n s i t y f u n c t i o n s , i The second c o n d i t i o n makes i t p o s s i b l e to r e q u i r e that W pre s e r v e s unanimity. However, t h i s c o m p a r a b i l i t y assumption i s subje c t to some i n t e r p r e t a t i o n . For i n s t a n c e , the u 's might be i taken t o be e i t h e r s c a l e comparable or f u l l y comparable (Sen 1970, p. 106), depending on whether we are w i l l i n g to p o s t u l a t e that the transformed u t i l i t y f u n c t i o n s , v = au + b , are i i i comparable f o r a l l a > 0 and b e R or onl y when b = b f o r a l l i i 1<i<n, r e s p e c t i v e l y . Given that the u t i l i t i e s are f u l l y comparable, say, A c z e l ' s four p o s t u l a t e s (Axioms A-D) are e a s i l y i n t e r p r e t a b l e and appear to be e n t i r e l y compatible with our i n t u i t i v e e x p e c t a t i o n s c oncerning the behaviour of a "reasonable" s o c i a l - w e l f a r e f u n c t i o n . By Theorem 3.2.3, the j o i n t u t i l i t y u = W(u 1 f...,u ) n 1 12 should thus be given by n u = ir 1 [ Z w yp(u ) ] i=1 i i f o r some continuous and s t r i c t l y i n c r e a s i n g f u n c t i o n ^ : ( 0 , » ) — > n R and w,,...,w £ 0, I w = 1 . Furthermore, i t would seem n i = 1 i n a t u r a l to demand that our p o o l i n g operator W obeys w(au,+b,...,au +b) = a«W(u,,...,u ) + b (3.2.7) n n f o r a l l a>0 and beR such that au +b > 0 on the whole domain of i u ,i=1,...,n. T h i s i n v a r i a n c e p r o p e r t y of W guarantees that no i dilemma w i l l a r i s e from i n t e r c h a n g i n g the o p e r a t i o n s of p o o l i n g and t r a n s f o r m i n g the s c a l e s of b e l i e f . As i t turns out, t h i s e x t r a requirement i s s u f f i c i e n t to imply a p r e c i s e form for w. P r o p o s i t i o n 3.2.5 The l i n e a r o p i n i o n pool i s the only q u a s i - a r i t h m e t i c p o o l i n g operator which s a t i s f i e s Equation (3.2.7) f o r a l l a and b i n (0,»). P r o o f : I t i s w e l l known (Theorem 84; Hardy, L i t t l e w o o d & P61ya 1934, p. 68) that the only q u a s i - a r i t h m e t i c means M(x 1 r...,x ) which n 113 s a t i s f y M(ax,,...,ax ) = a«M ( x 1 r . . . , x ) f o r a l l x 1 f . . . , x , a>0 n n n are the weighted means of order a , n a 1 / a [ I w ( i ) x ] , a * 0 ; i=1 i n w ( i ) I I x , a=0, i = 1 i M (x , . . .,x ) = a 1 n which we have a l r e a d y encountered i n Chapter 2 f o r 0<a<1. Of those, only M, obeys the second c o n d i t i o n M (x,+b,...,x +b) = a n M (x,,...,x )+b f o r a l l x,,...,x ,b > 0. a n n In the language of p o o l i n g o p e r a t o r s , t h i s means that n n W(u,,...,u ) = I w u f o r some w £ 0 with Z w =1 whenever W n i=1 i i i i=1 i i s q u a s i - a r i t h m e t i c and C o n d i t i o n (3.2.7) h o l d s . • A s i m i l a r argument c o u l d a l s o be used i f the u 's were only i assumed to be s c a l e comparable. In that case, W would have the added bonus to s a t i s f y W(au,+b 1 f ,au +b ) = a « W ( u 1 f — , u ) + b n n n n f o r some b = I w b , b > 0. I t may be of some i n t e r e s t to note i=1 i i that the only q u a s i - a r i t h m e t i c p o o l i n g o p e r a t o r s , W, which s a t i s f y the f o l l o w i n g g e n e r a l i z a t i o n of (3.2.7) are d i c t a t o r s h i p s ; 1 14 W(a,u,+b,,...,a u +b ) = n n n W(a 1 r...,a )«W(u 1 f...,u ) + W(b,,...,b ). n n n If we t h i n k of a "proper" s o c i a l - w e l f a r e f u n c t i o n as one which takes account of the p r e f e r e n c e p a t t e r n s of each of the i n d i v i d u a l s concerned, then t h i s l a s t o b s e r v a t i o n c o u l d be i n t e r p r e t e d as saying that such a f u n c t i o n w i l l not e x i s t u n l e s s i n t e r p e r s o n a l comparisons of u t i l i t y a re p o s s i b l e . Example 3.1.2 (continued) Suppose that p 1 f . . . , p e II represent the n o p i n i o n s of a n group of experts and that e U are t h e i r r e s p e c t i v e n ( s u b j e c t i v e ) l i k e l i h o o d s f o r 0 given that a s i n g l e d a t a - s e t , D = {X,,...,X }, has been observed by a l l the e x p e r t s . We s h a l l s assume that upon ob s e r v i n g D, each expert updates h i s b e l i e f s i n accordance with the f o l l o w i n g r u l e : q $ «p , i=1,...,n, (3.2.8) i i i where q i s the o p i n i o n of the i - t h expert given D. T h i s i formula i s i m p l i e d by Bayes' Theorem i n the case of P - f u n c t i o n s : the n o r m a l i z a t i o n constant i s i r r e l e v a n t as p r o p e n s i t y f u n c t i o n s 1 15 a r e t o be t r e a t e d -and sometimes even i n t e r p r e t e d t h r o u g h c o n s i d e r a t i o n o f - b e t t i n g odds. Moreover, t h i s c o n s t a n t may not e x i s t i f p i s s u f f i c i e n t l y i mproper. So i t i s o m i t t e d and q i i i s r e g a r d e d as a l a b e l f o r the e q u i v a l e n c e c l a s s of a l l c o n s t a n t m u l t i p l e s of p (Novick & H a l l 1965, pp. 1105-1106). i The n a t u r a l c o u n t e r p a r t of an i s s u e r a i s e d i n S e c t i o n 2.4 a r i s e s h e r e : s h o u l d P - f u n c t i o n s be combined b e f o r e or a f t e r the o b s e r v a t i o n of the sample e v i d e n c e D. Note t h a t i f i t i s d e c i d e d t o p o o l f i r s t , t he d i s c r e p a n c y between the l i k e l i h o o d f u n c t i o n s of the e x p e r t s w i l l have t o be r e s o l v e d b e f o r e the j o i n t P - f u n c t i o n can be updated. The i d e a , h e r e , i s t o p o o l the l i k e l i h o o d s , and s i n c e t h e y a r e but o t h e r e x p r e s s i o n s of o p i n i o n ( P - f u n c t i o n s ) from the same e x p e r t s , i t would seem n a t u r a l t o use the same p o o l i n g f o r m u l a as f o r the p r i o r s . Moreover, i t would be d e s i r a b l e t h a t t h e o p e r a t i o n s of p o o l i n g and u p d a t i n g commute. An o p e r a t o r which does t h i s w i l l be c a l l e d " p r i o r - t o -p o s t e r i o r c o h e r e n t " a f t e r Weerahandi & Zidek (1978) who used i t as a s u b s t i t u t e f o r Madansky's term " e x t e r n a l B a y e s i a n i t y ; " i n t h i s t h e s i s , the two e x p r e s s i o n s a r e now v e s t e d w i t h d i f f e r e n t meanings. D e f i n i t i o n 3.2.6 n We say t h a t a p o o l i n g o p e r a t o r T:II — > I I i s p r i o r - t o - p o s t e r i o r  c o h e r e n t i f f T ( * , p , , . . . ,* p ) = T ( $ 1 , . . . , $ )«T(p,,...,p ) (3.2.9) n n n n 116 f o r a l l * ,p e II, i = 1,...,n. i i Note that t h i s d e f i n i t i o n does not i n v o l v e l o c a l i t y . Assuming that ( 3 . 2 . 8 ) h o l d s , p r i o r - t o - p o s t e r i o r coherence i s an independent c r i t e r i o n f o r s e l e c t i n g a p o o l i n g o p e r a t o r ; thus, i n theory at l e a s t , our search f o r such o p e r a t o r s c o u l d extend to n a l l a p p l i c a t i o n s T:I1 —>n. However, arguments given above suggest that p o o l i n g formulas should be l o c a l and s a t i s f y A c z e l ' s four p o s t u l a t e s , i . e . be q u a s i - a r i t h m e t i c p o o l i n g o p e r a t o r s . In that case, i t i s easy to see that Property ( 3 . 2 . 9 ) s i n g l e s out the l o g a r i t h m i c o p i n i o n p o o l . In f a c t , we w i l l show more: P r o p o s i t i o n 3 . 2 . 7 n Let T:II —>n be a l o c a l p o o l i n g operator which p r e s e r v e s the o r d e r i n g of b e l i e f s (POB) and i s p r i o r - t o - p o s t e r i o r coherent. There e x i s t w(1),...,w(n) > 0 such that n w ( i ) T(p,,...,p )(e) = n [p (e)] ( 3 . 2 . 1 0 ) n i = 1 i n f o r a l l 0e6 and (p,,...,p ) e U , i . e . T i s a l o g a r i t h m i c n n o p i n i o n p o o l . Moreover, Z w(i) = 1 whenever T p r e s e r v e s i=1 unanimity. 1 17 Proof: Write T(p,,...,p )(0) = G(p,(0),...,p (0)) f o r a l l 0e0 and n n P i , . . . , P e n . Using Equation (3.2.9), we see th a t G(x»y) = n n G(x)»G(y) f o r a l l x and y i n (0,») , and i t f o l l o w s from D e f i n i t i o n 3.1.7 that G i s s t r i c t l y i n c r e a s i n g i n each of i t s n v a r i a b l e s . Appealing t o Lemma 2.1.3, we conclude that n w ( i ) G(x,,...,x ) = II x f o r some w(i)>0, i = 1,...,n. The sum n i = 1 i n Z w(i) i s a r b i t r a r y , u n l e s s we r e q u i r e that G(x,...,x)=x f o r i=1 a l l x>0, i . e . T s a t i s f i e s the Unanimity P r i n c i p l e . • To i l l u s t r a t e t h i s r e s u l t , suppose t h a t a d e c i s i o n maker has c o l l e c t e d some data x,,...,x and that he regards the n l i k e l i h o o d f u n c t i o n $ (0) which each of these items p r o v i d e s as i an i n d i v i d u a l expert o p i n i o n . Formula (3.2.10) or perhaps n 1/n $ = n [ $ (0)] (3.2.11) i = 1 i might then be used to o b t a i n the r e p r e s e n t a t i v e l i k e l i h o o d , A l t e r n a t i v e l y , i f the sample i s taken as a whole, there w i l l be j u s t n=1 l i k e l i h o o d . Both (3.2.10) and (3.2.11) would then r e t u r n t h i s l i k e l i h o o d i n a p o s s i b l y r e n ormalized form. T h i s r e s u l t would, i n g e n e r a l , d i f f e r from that of Equation (3.2.11). 118 The p o i n t , however, i s t h a t the p o o l i n g operator i s not i n d i c a t i n g , i n any given c o n t e x t , which p r o p e n s i t y f u n c t i o n s should be combined, but r a t h e r i t i s p r o v i d i n g a means of p o o l i n g such f u n c t i o n s once they have been s e l e c t e d . I n c i d e n t a l l y , i f the data presented i n the l a s t paragraph are obtained from independent measurements and t h e i r j o i n t l i k e l i h o o d i s found by f i r s t computing the j o i n t sampling d i s t r i b u t i o n and then i n v e r t i n g t h i s i n the usual way, the n n 1 /n r e s u l t w i l l be II * . T h i s d i f f e r s from the q u a n t i t y [ II 4> ] i=1 i i=1 i which would be obtained from Equation (3.2.11). The l a t t e r i s j u s t the r e s c a l e d v e r s i o n of the former and, i f n—>», the Strong Law of Large Numbers i m p l i e s t h a t i t converges to a constant m u l t i p l e of e x p { l ( f ,f )}, where I denotes the K u l l b a c k - L e i b l e r 0 0 O d i s c r i m i n a t i o n measure, * (0) = f ( x |0), f(«|0) i s the sampling i i d e n s i t y i f 0 i s the "true s t a t e of nature," and 0 O i s the true r e a l i z a t i o n of the random v a r i a b l e 0. I f , on the other hand, the data are h i g h l y dependent, say x = x,, i=2,...,n, then i Equation (3.2.11) would g i v e , or very n e a r l y g i v e , the j o i n t l i k e l i h o o d i t s e l f . In S e c t i o n 3.3 below, we s h a l l address the problem of f i n d i n g n o n - l o c a l p r i o r - t o - p o s t e r i o r coherent p o o l i n g o p e r a t o r s . Let us mention i n p a s s i n g t h a t a c h a r a c t e r i z a t i o n of the weighted mean of order a (M d e f i n e d above) o b t a i n s i f the a 119 v a l i d i t y of C o n d i t i o n (3.2.9) i s l i m i t e d t o those cases where there was mutual agreement a p r i o r i on the l i k e l i h o o d 4> = • • • ^ • n P r o p o s i t i o n 3.2.8 n Let T:I1 —>n be a q u a s i - a r i t h m e t i c p o o l i n g o p e r a t o r . I f T s a t i s f i e s T($p,,...,$p ) = ••T(p,,...,p ) n n fo r a l l $, p,,...,p e n, then T = M f o r some a e (0,»). n a Proof: T h i s i s because the weighted means of order a>0 are the n only q u a s i - a r i t h m e t i c means M:(0,») — > ( 0 , » ) which are "homogeneous." See Theorem 84 on page 68 i n Hardy, L i t t l e w o o d & P61ya (1934). • Example 3.1.3 (continued) In t h i s case, the p o o l i n g operator P ( L 1 f . . . , L ) i s n a u t o m a t i c a l l y l o c a l , because 0 = {H0} i s a s i n g l e t o n . The i n t e r p r e t a t i o n of A c z e l ' s axioms causes no d i f f i c u l t y e i t h e r . For i n s t a n c e , Axiom B expresses the evident requirement that a set S, of P-values should be more s i g n i f i c a n t , as a whole, than another set S 2 i f the P-values i n S, are s m a l l e r than the corresp o n d i n g P-values i n S 2. As another example, the 120 i n e q u a l i t y min{L |i=1,...,n} < P(L,,...,L ) ^ max{L |i=1,...,n} i n i (a consequence of Axioms A and B taken together) accounts f o r the f a c t that because the data upon which J , , . . . , J are based n cannot be combined d i r e c t l y ( e i t h e r because they are u n a v a i l a b l e or incomparable due to d i f f e r e n c e s i n the q u a n t i t a t i v e or q u a l i t a t i v e a spects of the v a r i o u s d e s i g n s ) , we do not expect the combination t e s t to give us more ( l e s s ) c onfidence i n H 0 than the most ( l e a s t ) o p t i m i s t i c of the observed l e v e l s L . i Theorem 3.2.3 suggests that we use the t e s t s t a t i s t i c n P(L,,... ,L ) = \p'1 [ Z w ^ ( L ) ], n i=1 i i where i//:(0,°»)—>R would be continuous and s t r i c t l y i n c r e a s i n g on i t s domain. Moreover, u s i n g the f a c t that i n g e n e r a l any s t r i c t l y i n c r e a s i n g continuous t r a n s f o r m a t i o n x(s) of a s t a t i s t i c S w i l l produce the same one-sided t e s t as S ( c f . L i p t a k 1958, p. 176), we can r e s t r i c t our a t t e n t i o n to n P[\p] (L, , . . . ,L ) = Z w v//(L ) (3.2.12) n i=1 i i n with s u i t a b l e weights w £0, Z w = 1 , say. T h i s f a m i l y of t e s t i i = 1 i s t a t i s t i c s was f i r s t i n t r o d u c e d by L i p t a k (1958) and comprises 121 (i) Good's ( 1 9 5 5 ) weighted version of Fisher's omnibus procedure [\/>(x) = log(x) ]; ( i i ) the so-called inverse normal procedure U " 1 (x) = JI (-»,x)-exp{ (-1/2) (y2+log(27r) }dy]; and ( i i i ) the more recent l o g i t s t a t i s t i c of Mudholkar & George (1979) [^(x)= log(x/( 1-x)) ]. In the two l a t t e r cases, the domain of i// i s r e s t r i c t e d to ( 0 , 1 ) , but t h i s can be j u s t i f i e d by appealing to a d i f f e r e n t form of Theorem 3.2.3 where the P-functions would only take values in [ 0 , 1 ] . The following result vindicates the use of the quasi-arithmetic weighted means in t h i s p a r t i c u l a r a p p l i c a t i o n . Theorem 3.2.9 (Liptak 1958) Each member P[^] of the class ( 3 . 2 . 1 2 ) y i e l d s a most powerful test against some s p e c i f i c a l t e r n a t i v e . Moreover, P[\p] i s an unbiased test for the sample consisting of the P-values whenever the o r i g i n a l test s t a t i s t i c s J 1 r . . . , J are. n Proof: A detailed proof of t h i s theorem i s contained in Liptak's paper. However, we would l i k e to mention that the f i r s t statement i s a d i r e c t consequence of the fact that any test s t a t i s t i c P[i£] of the form ( 3 . 2 . 1 2 ) s a t i s f i e s Birnbaum's ( 1954) "condition 1." • In general, the a l t e r n a t i v e against which a p a r t i c u l a r test P[i//] i s admissible may be quite obscure, so t h i s result does not constitute a strong basis for choosing one form of P[<//] 122 over another. Moreover, the a l t e r n a t i v e i s g e n e r a l l y s p e c i f i e d only vaguely. In h i s paper, L i p t a k proposed to circumvent t h i s d i f f i c u l t y by using \p = the i n v e r s e of the cumulative d i s t r i b u t i o n f o r the standard normal N(0,1), i . e . the " i n v e r s e normal procedure." He claimed -without p r o o f - that t h i s p a r t i c u l a r c h o i c e , a p a r t from being convenient from a computational p o i n t of view, was optimal f o r a l a r g e c l a s s of one-sided hypothesis t e s t i n g problems, i n c l u d i n g those where the p o s s i b l e d i s t r i b u t i o n s are generated by d e n s i t i e s belonging to the e x p o n e n t i a l f a m i l y . More r e c e n t l y , Scholz (1981) proposed to l e t the P-values themselves choose the "proper" f u n c t i o n v/> by t a k i n g t h a t 4> ( s u i t a b l y s t a n d a r d i z e d ) which y i e l d s the l a r g e s t p o s s i b l e value of P[<//]. He d e s c r i b e s h i s p r o p o s a l as an a p p l i c a t i o n of Roy's (1953) u n i o n - i n t e r s e c t i o n p r i n c i p l e to a nonparametric s e t t i n g . Another way of comparing the P [ ^ ] ' s would be to compute t h e i r Bahadur (1967) e f f i c i e n c y , i . e . the l i m i t i n g r a t i o of sample s i z e s r e q u i r e d by any two given t e s t s t a t i s t i c s of the form (3.2.12) to a t t a i n e q u a l l y small s i g n i f i c a n c e l e v e l s . Thus, L i t t e l l & F o l k s (1971) showed that F i s h e r ' s method i s always at l e a s t as e f f i c i e n t , i n the Bahadur sense, as three other well-known competing methods, i n c l u d i n g L i p t a k ' s i n v e r s e normal procedure. In f a c t , a s tronger r e s u l t of the same authors has the f o l l o w i n g consequence (we are g r a t e f u l to Dr. A. John Petkau f o r b r i n g i n g t h i s second paper of L i t t e l l & F o l k s to our a t t e n t i o n ) : 123 Theorem 3 . 2 . 1 0 ( L i t t e l l & F o l k s 1973) Let P(L,,...,L ) be any t e s t s t a t i s t i c which preserves the n o r d e r i n g of b e l i e f s (POB) concerning the v a l i d i t y of the n u l l h y p o t h e s i s (Axiom B). Then P i s at most as e f f i c i e n t , i n the Bahadur sense, as F i s h e r ' s omnibus procedure. Proof; I f P(L,,...,L ) s a t i s f i e s Axiom B, then the s t a t i s t i c n S ( J , , . . . , J ) = -P(L,,...,L ) which r e j e c t s the n u l l h y p o t h e s i s n n f o r l a r g e v a l u e s of S s a t i s f i e s the c o n d i t i o n of the theorem which appears on page 193 i n L i t t e l l & F o l k s ( 1 9 7 3 ) . • Remark 3 .2 .11 I t i s not too d i f f i c u l t to f i n d continuous and s t r i c t l y i n c r e a s i n g f u n c t i o n s i / / : (0,1)—>R f o r which the t e s t s t a t i s t i c n I w(i) i£(L ) w i l l have the same Bahadur e f f i c i e n c y as the i = 1 i n w ( i ) corr e s p o n d i n g weighted F i s h e r procedure n L . For example, i = 1 i one c o u l d take ^ to be the i n v e r s e of the cumulative d i s t r i b u t i o n of a gamma, an inverse-Gaussian or a Laplace random v a r i a b l e . However, Theorem 3 . 2 . 1 0 a s s e r t s that no t e s t based on a s t a t i s t i c of the form ( 3 . 2 . 1 2 ) w i l l surpass the omnibus procedure. In summary, a set of weak and a p p e a l i n g c o n d i t i o n s was developed which was shown to c h a r a c t e r i z e the q u a s i - a r i t h m e t i c 124 p o o l i n g o p e r a t o r s (3.2.1). I t was assumed, fundamentally, that the s c a l e s of b e l i e f which were used by the v a r i o u s experts to express t h e i r o p i n i o n s were intercomparable, so t h a t the p o o l i n g o p e r a t o r s c o u l d be l e g i t i m a t e l y supposed to be unanimity p r e s e r v i n g . A l o c a l i t y assumption was a l s o made f o r mathematical convenience. Furthermore, we saw t h a t , depending on the a p p l i c a t i o n , e x t r a i n v a r i a n c e c o n d i t i o n s p e r t a i n i n g to the s c a l e s of b e l i e f can be imposed to reduce the c l a s s of a c c e p t a b l e o p e r a t o r s . However, c a r e must be taken i n imposing such c o n d i t i o n s , as the consequent r e d u c t i o n s may sometimes r u l e out " o p t i m a l " s o l u t i o n s . In the f o l l o w i n g s e c t i o n , we take a look a t the more c h a l l e n g i n g problem of p o o l i n g P - f u n c t i o n s whose s c a l e s of b e l i e f are not n e c e s s a r i l y comparable. 3.3 D e r i v i n g the l o g a r i t h m i c o p i n i o n pool I t i s shown i n t h i s s e c t i o n that the s o - c a l l e d general l o g a r i t h m i c o p i n i o n p o o l , L, i s a reasonable c h o i c e of a P-f u n c t i o n p o o l i n g operator even when degrees of b e l i e f are not intercomparable. When © i s f i n i t e , another product formula i s d e r i v e d which i s p r i o r - t o - p o s t e r i o r coherent and p r e s e r v e s the o r d e r i n g of b e l i e f s . At the end of the s e c t i o n , a p a r a l l e l i s drawn between our approach to p o o l i n g P-func.t.iojns and Nash's (1950) s o l u t i o n to the m u l t i - p e r s o n c o o p e r a t i v e d e c i s i o n problem. 125 D e f i n i t i o n 3.3.1 The g e n e r a l l o g a r i t h m i c o p i n i o n pool i s d e f i n e d by n w ( i ) L ( p 1 f . . . , p ) = C(p,,...,p )• n [p (0)] (3.3.1) n n i=1 i n f o r a l l p,,...,p e n and 0e0, where C:I1 — > ( 0 , » ) i s some n u n s p e c i f i e d f u n c t i o n and w(1),...,w(n) are non-negative n c o n s t a n t s such that Z w(i) > 0. i = 1 The operator L d e f i n e d above i s not l o c a l because the f u n c t i o n C depends on ( p 1 f . . . , p ). If C ( p 1 r . . . , p ) = 1 f o r a l l n n c h o i c e s of p 1 f . . . , p e II, then L reduces t o the l o g a r i t h m i c pool n (3.2.10). In a n t i c i p a t i o n of the developments below, we make the f o l l o w i n g D e f i n i t i o n 3.3.2 n n The r e l a t i v e p r o p e n s i t y mapping i s a f u n c t i o n RP:II x © 2 — > ( 0 , » ) n which maps any (n+2)-tuple ( p 1 f . . . , p ,0,TJ) i n II x© 2 to the n v e c t o r of q u o t i e n t s RP(p,,...,p ,0,7?) = (p, ( 0)/p, (TJ) , . . . ,p (0)/p ( T J ) ) . n n n It i s immediate t h a t the a p p l i c a t i o n RP induces an n n equ i v a l e n c e r e l a t i o n on D = II x 0 2 . I f two elements of II x © 2 , 126 say CIT and d 2 , are c a l l e d RP-equivalent whenever RP(d,) = R P ( d 2 ) , then D may be decomposed i n t o RP-equivalence c l a s s e s n o b t a i n e d through RP's i n v e r s e mapping. The set (0,») may be regarded as a l a b e l s et f o r the q u o t i e n t space D/RP. As w i l l be shown, the f o l l o w i n g p r o p e r t y c h a r a c t e r i z e s the general l o g a r i t h m i c o p i n i o n p o o l . D e f i n i t i o n 3.3.3 n We say that a p o o l i n g operator T : I I —>n i s r e l a t i v e p r o p e n s i t y  c o n s i s t e n t (RP-C) i f f T(p,,...,p ){6) T ( q 1 , . . . , q )(TJ) (3.3.2) n > n T(p!,...,p ) ( X ) T(q,,...,q ) U) n n whenever RP(p,,...,p , 0 , X ) > RP(q,,...,q , 7 ? , £ ) , Pi,--«»P , n n n q 1 f . . . , q being a r b i t r a r y elements of n and 0,T ? , X,£ belonging to n 0. To i n t e r p r e t t h i s new concept, i t i s u s e f u l to decompose C o n d i t i o n (3.3.2) i n t o two p a r t s , namely ( i ) T ( p p )(c9) T(p,,...,p )(r?) n > n T(p,,...,p ) ( X ) T(p,,...,p ) ( I ) n n whenever RP(p 1 f...,p , 0 , X ) > RP(p 1 r...,p , T J , ^ ) ; n n and ( i i ) T(p,,...,p )(d) T(q,,...,q ){6) n = n T(p,,...,p ) ( X ) T(q!,...,q ) ( X ) n n 127 whenever RP(p,,...,p ,0,X) = RP(q,,...,q ,0,X). n n I t i s easy to check that C o n d i t i o n s ( i ) and ( i i ) together are e q u i v a l e n t to (3.3.2). The f i r s t of these c o n d i t i o n s says that a good p o o l i n g procedure should p r e s e r v e any p r i o r consensus of the form "the odds i n favour of the occurrence of 0 versus X are b e t t e r than the odds f o r TJ versus £." In p a r t i c u l a r , note that any operator which s a t i s f i e s t h i s requirement w i l l a u t o m a t i c a l l y p r e s e r v e the o r d e r i n g of b e l i e f s , i n the sense of D e f i n i t i o n 3.1.7, and -by way of consequence-the " c o n s i s t e n c y " c o n d i t i o n which appears i n Lemma 3.1.6. Here again, C o n d i t i o n ( i i ) i s a s i m p l i f y i n g assumption; only t h i s time i t i n v o l v e s o d d s - r a t i o s . I t may be i n t e r p r e t e d i n the same way as the C o n d i t i o n ( i i ) which appears i n S e c t i o n 3.1. We are now i n a p o s i t i o n to s t a t e and prove the p r i n c i p a l r e s u l t of t h i s s e c t i o n . Theorem 3.3.4 Suppose that © c o n t a i n s at l e a s t t h r e e d i s t i n c t elements. The gene r a l l o g a r i t h m i c o p i n i o n p o o l , L, i s the only r e l a t i v e p r o p e n s i t y c o n s i s t e n t p o o l i n g o p e r a t o r . Proof: I f T i s any P - f u n c t i o n p o o l i n g operator s a t i s f y i n g the hypotheses of the theorem, C o n d i t i o n (3.3.2) above has the immediate i m p l i c a t i o n t h a t the f u n c t i o n Q ( p i , . . . , p ,#,»?) = n 1 2 8 T ( p 1 f . . . , p , 0 ) / T ( p , , . . . , p ,T?) must be c o n s t a n t on R P - e q u i v a l e n c e n n n c l a s s e s of D. T h e r e f o r e , t h e r e e x i s t s a mapping H : ( 0 , » ) — > ( 0 , » ) such t h a t Q=HnRP, t h e symbol n r e p r e s e n t i n g as b e f o r e the c o m p o s i t i o n of f u n c t i o n s . Q — > ( 0 , » ) RP 7» / / / / / H / / • / n ( 0 , » ) P i c k 0 , 7 ? , X , t h r e e d i s t i n c t elements of 6, and l e t x and y be two n a r b i t r a r y v e c t o r s i n ( 0 , ° ° ) . I f p,,...,p e n a r e chosen so n t h a t p ( 0 ) = x , p (T?) = 1/y and p ( X ) = 1 f o r a l l 1£i£n, then i i i i i H(x-y) = H ( p , ( 0 ) / p , ( i j ) , . . . , p ( 0 ) / p (T?)) = Q(p,,...,p , 0 , T ? ) = n n n Q ( p i , . . . , P > 8 ,\) «Q(pi,... ,p , X , T ? ) = H(p, ( 0 ) , ... ,p ( 0 ) ) « n n n H( 1/p, (T?) , ... , 1/p (T?)) = H ( 5 f)-H(y). I t a l s o f o l l o w s from the n RP-C c o n d i t i o n t h a t H i s n o n - d e c r e a s i n g i n each of i t s n n w ( i ) v a r i a b l e s , so t h a t H(x) = n x w i t h some f i x e d numbers i = 1 i w(1),...,w(n) > 0 by an a p p l i c a t i o n of Lemma 2 . 1 . 3 ( h e r e , POB p l a y s i t s r o l e as a r e g u l a r i t y c o n d i t i o n ) . n w( i ) C o n s e q u e n t l y , Q ( p i , . . . , p ,0,T?) = II [p (0)/p (TJ)] f o r a l l n i=1 i i 1 29 p 1 f . . . , p e IT and 0,TJ e G, i . e . we have shown t h a t the f u n c t i o n n n w ( i ) C(p,,...,p )(0) = T(p,,...,p )(8)/ n [p (0)] n n i=1 i i s independent of 0! • T h i s theorem i s not t r u e i f 0 c o n t a i n s e x a c t l y two elements, as the f o l l o w i n g counter-example shows. Example 3.3.5 n L e t 0 = {0,r?} and d e f i n e T:II —>n by T(p,, ,p ) = (p,+p 2)/2. n Then T i s r e l a t i v e p r o p e n s i t y c o n s i s t e n t and even unanimity p r e s e r v i n g , but c l e a r l y T * t. As before, the problem of choosing the weights w(i) remains and i s not addressed here. Note a l s o that the f u n c t i o n C i n Equation (3.3.1) i s undetermined, except f o r the t r i v i a l requirement C(p,...,p) = 1 f o r a l l p e n i f L s a t i s f i e s the Unanimity P r i n c i p l e (and the ex p e r t s ' s c a l e s of b e l i e f are i n t e r c o m p a r a b l e ) . At the moment, i t i s not c l e a r to us what r o l e t h i s f u n c t i o n p l a y s or even how i t c o u l d be i n t e r p r e t e d . I f we i n s i s t that L should be p r i o r - t o - p o s t e r i o r coherent i n the n sense of D e f i n i t i o n 3.2.6, i t i s necessary to have L w(i) = 1 i=1 and a l s o C ( p , , — , p ) ' C ( q 1 f . . . , q ) = C(p,«q,,...,p «q ) f o r n n n n n a l l ( p 1 f . . . , p ) , ( q i , . . . , q ) e n ; but even that requirement i s n n 130 not strong enough to completely determine C. The f o l l o w i n g p a r t i a l r e s u l t g i v e s s t i l l another i n d i c a t i o n of the l a r g e v a r i e t y of p o o l i n g o p e r a t o r s which i s encompassed by the term "non-local.'' P r o p o s i t i o n 3.3.6 n Let 0 = {0,,...,0 } be f i n i t e , and assume that T:I1 —>n i s a POB m p o o l i n g operator which i s p r i o r - t o - p o s t e r i o r coherent. There e x i s t s a set {w(i,j,k)|1<i<n,1£j,k^m} of p o s i t i v e c o n s t a n t s such t h a t m n w ( i , j , k ) T(p,,...,p )(0 ) = n n [p (0 )] n k j=1i=1 i j f o r a l l p,,...,p e II and ke{ 1 ,... ,m}. n Proof: F i x 0 e 0 and c o n s i d e r H(p,,...,p ) = T(p,,...,p )(0 ) as a k n n k n group homomorphism between n and (0,») with m u l t i p l i c a t i o n . n Then H ( p 1 f . . . , p ) = n H (p ) where H :I1—>(0,») i s d e f i n e d by n i=1 i i i H (p ) = H(1,...,p ,...,1) f o r each i=1,...,n, and we have i i i H (p«q) = H (p)«H (q) whenever p and q belong to II. Each H can i i i i m be decomposed f u r t h e r as H (p) = n H (p(0 )) where H : i j»1 i j D i j 131 (0,»)—>(0,») i s d e f i n e d by H (x) = H ( x ) , where xell i s that i j i P - f u n c t i o n whose value at 8 i s x and 1 otherwise. j Now, H i s a homomorphism on (0,»), and i t i s non-decreasing i j because T p r e s e r v e s the o r d e r i n g of b e l i e f s . Using Lemma 2.1.3, w ( i , j , k ) i t f o l l o w s t h a t H (x) = x f o r some w ( i , j , k ) > 0, with i j k i n d i c a t i n g a p o s s i b l e dependence on 8 . Combining a l l these k f a c t s , we o b t a i n the d e s i r e d c o n c l u s i o n . • To conclude t h i s s e c t i o n , we would l i k e to draw a p a r a l l e l between the general l o g a r i t h m i c o p i n i o n pool and Nash's (1950) s o l u t i o n to the s o - c a l l e d " b a r g a i n i n g problem." The two are r e l a t e d i n a manner which w i l l now be d e s c r i b e d . In g e n e r a l , given an a c t i o n space A and a space of randomized d e c i s i o n r u l e s D*, l e t u (_) = J7u (a , 0)S(da)b (8)u(68), i i i where u denotes the i - t h p l a y e r ' s u t i l i t y f u n c t i o n and b i s a i i p r i o r or p o s t e r i o r d i s t r i b u t i o n , whichever i s a p p r o p r i a t e . Then Nash's axioms imply that the s o l u t i o n s , £*, are those which n 1/n maximize the symmetric product n [u ($)-c ] where c i=1 i i i 132 denotes the i - t h p l a y e r ' s s t a t u s quo p o i n t , i . e . the amount i n u t i l i t y which he w i l l have i n the event that the group f a i l s t o agree on a c h o i c e f o r £. T h i s maximization i s s u b j e c t to u ($) i > c f o r a l l i=1,...,n (see Weerahandi & Zidek 1981 f o r f u r t h e r i d e t a i l s ) . Now suppose 0 = {6i,...,6 } i s f i n i t e , as i n P r o p o s i t i o n m m 3.3.6. Assume f u r t h e r that u (a,0) = 6(a,0) + c / £ b (0 ) i i j=1 i j where 6(a,0) i s the Kronecker d e l t a f u n c t i o n . Then the Nash s o l u t i o n maximizes n 1/n n 1/n n [/J6(a,0)$(da)b ( 0 M d 0 ) ] = n [Jb (a)$(da)] i=1 i i=1 i where v denotes the usual counting measure. Observe that i f $ i s r e s t r i c t e d to be nonrandomized, the o p t i m a l c h o i c e of a i s n 1/n the 0 which maximizes II [b (0)] , e s s e n t i a l l y the q u a n t i t y i = 1 i which would be obtained from Equation (3.3.1) with w(1) = ... = w(n) = 1/n. T h i s o b s e r v a t i o n lends some a d d i t i o n a l support to the g e n e r a l l o g a r i t h m i c p o o l i n g r e c i p e . C l e a r l y , a s i m i l a r argument c o u l d be found f o r Stone's l i n e a r p o o l i n g operator by a p p e a l i n g to the work of Bacharach (1975) which i n t u r n r e l i e s on an unpublished c o n t r i b u t i o n of 133 Madansky. T h i s work i n c l u d e s a theorem which shows that the n optimal d e c i s i o n r u l e maximizes Z w u ( $ ) . Indeed, the s o r t of i=l i i r e d u c t i o n which i s sketched i n the l a s t paragraph would y i e l d the l i n e a r o p i n i o n p o o l . I t should be noted, however, that Bacharach's r e s u l t i m p l i c i t l y assumes the i n t e r c o m p a r a b i l i t y of u t i l i t i e s . T h i s i m p l i c i t h y p o t h e s i s a r i s e s when a theorem of B l a c k w e l l & G i r s h i c k (1954) i s invoked i n p r o v i n g the a s s e r t e d c o n c l u s i o n . The B l a c k w e l l - G i r s h i c k r e s u l t d e a l s with the c l a s s i c a l d e c i s i o n problem where the i ' s represent d i f f e r e n t s t a t e s of nature, not d i f f e r e n t p l a y e r s . In that s i t u a t i o n , there i s only one p l a y e r and presumably he would have no d i f f i c u l t y comparing h i s own p r e f e r e n c e s and hence deducing the u t i l i t y f u n c t i o n s f o r the d i f f e r e n t s t a t e s of nature, i . In c e r t a i n d e c i s i o n or e s t i m a t i o n problems, a p r o p e n s i t y f u n c t i o n may be used to f i n d the 6 i n 0 f o r which there i s the maximum j o i n t p r o p e n s i t y (MJP). I f , f o r example, b {&) = i exp{0u - A(6)}, the 8 of MJP i s the unique s o l u t i o n of A'(0) = i _ n u = I a u . T h i s has the c u r i o u s consequence p o i n t e d out i n i = 1 i i Weerahandi & Zidek (1978) that i f the u 's are widely separated, i ix may w e l l have a very low p r o p e n s i t y as measured by the i n d i v i d u a l P - f u n c t i o n s , b . The corres p o n d i n g d i f f i c u l t y with i the l i n e a r o p i n i o n pool i s t h a t the j o i n t p r o p e n s i t y f u n c t i o n i n 134 t h i s case i s multi-modal. In p l a c e of a s i n g l e r e p r e s e n t a t i v e u (say M) of low p r o p e n s i t y r e l a t i v e t o each b , a f a m i l y of i n o n r e p r e s e n t a t i v e M ' S (approximately the u 's themselves) i s i o b tained, each of h i g h p r o p e n s i t y r e l a t i v e to e x a c t l y one of these b 's. i By extending the domain of the p o o l i n g operator from 0 to D* i n the manner advocated by Weerahandi & Zidek (1978), the anomalies d e s c r i b e d i n the p r e v i o u s paragraph can e a s i l y be circumvented. In t h i s e x t e n s i o n , which i s suggested by the Nash theory sketched above, b (0) i s r e p l a c e d by b ($) = J*b (0)$(d0) i i i and the 0 of MJP i s r e p l a c e d by the $ of MJP; i n the s i t u a t i o n we have j u s t d e s c r i b e d , t h i s would l e a d to a randomized c h o i c e amongst, approximately, the widely separated M ' S . The v a r i o u s i p o o l i n g o p e r a t o r s suggested i n t h i s chapter would remain unchanged under t h i s e x t e n s i o n of domain. In f a c t , t h e i r d e r i v a t i o n would go through i n e x a c t l y the same way, s i n c e the b ($)'s are P - f u n c t i o n s too! i F i n a l l y , i t should be added that we have not attempted to extend our d e f i n i t i o n of p r o p e n s i t y f u n c t i o n to comprise those whose range i n c l u d e s z e r o . Most d e r i v a t i o n s of t h i s chapter would run i n t o d i f f i c u l t y i n t h i s case. Again, the problem we face i n t h i s s i t u a t i o n i s not u n l i k e t h a t encountered i n 135 c o n v e n t i o n a l Bayesian a n a l y s i s , when the p r i o r and the l i k e l i h o o d f u n c t i o n s have d i s j o i n t supports and some i m p r o v i s a t i o n i s i n o r d e r . 136 IV. SUGGESTIONS FOR FURTHER RESEARCH The present work has r e i n f o r c e d , i f a n y t h i n g , the view that there i s no "one best way" to aggregate e x p r e s s i o n s of b e l i e f . Along with Savage (1971), one c o u l d say that "what to do when do c t o r s d i s a g r e e has always been, and w i l l always be a quandary." However, there i s great p o t e n t i a l value i n the idea of choosing amongst the i n f i n i t e number of s y n t h e t i c experts that c o n s t i t u t e the convex c l o s u r e of a panel; witness the v a r i o u s s t u d i e s ( c f . , e.g., Sanders 1963; Winkler 1971; Shapiro et a l . 1977,1979) i n which composite d i s t r i b u t i o n s have been observed to p r e d i c t with g r e a t e r accuracy than most i n d i v i d u a l e x p e r t s . Granted, the evidence i s s t i l l mainly e m p i r i c a l and remains to be probed with sound a n a l y t i c a l models; but, says Hogarth (1977, p. 241), "there would seem to be l i t t l e doubt that the general r e s u l t s concerning the r e l i a b i l i t y and v a l i d i t y of average judgments in the form of p o i n t e s t i m a t e s . . . w i l l c a r r y over t o p r o b a b i l i t y d i s t r i b u t i o n s . " The development of such models i s one urgent task t h a t l i e s ahead of us. In the paper of Savage which i s c i t e d above, the author a r g u e s - a l b e i t i n d i r e c t l y - t h a t t h e c h r o n i c l a c k of dependable techniques of communication from which our s o c i e t y s u f f e r s makes the aggregation of o p i n i o n s d i f f i c u l t . T h i s 4 d i f f i c u l t y i s reduced at l e a s t i n the case which we d e s c r i b e d where i t i s s p e c i f i e d at the outset that the purpose of the d e c i s i o n maker i s " to l e a r n more" or perhaps to make a f o r e c a s t , but not to 137 take d e c i s i o n s . A l a c k of candor or i n e f f e c t i v e communication, gross exaggeration or e x c e s s i v e deference, f o r example, are more l i k e l y to be witnessed in s i t u a t i o n s where a group of experts f i n d s t h a t they are competing with one another f o r some s t r a t e g i c advantage. Moreover, even i n d e c i s i o n - o r i e n t e d t a s k s , one would expect such s o l i d l y e s t a b l i s h e d r u l e s as the l i n e a r or the l o g a r i t h m i c o p i n i o n p o o l s to show some robustness to "reasonable" departures from the assumption of n e a r l y i d e n t i c a l u t i l i t y f u n c t i o n s f o r the e x p e r t s . N e v e r t h e l e s s , we agree with Savage t h a t the need i s there to d e v i s e a p p r o p r i a t e methods of communication which w i l l h e l p a panel to h o n e s t l y share a l l r e l e v a n t f a c t u a l i n f o r m a t i o n and make i t p o s s i b l e f o r i t s members to a s s i s t one another i n t h i n k i n g t h e i r b e l i e f s through thoroughly. To a l a r g e degree, the problem of choosing an a p p r o p r i a t e scheme f o r weighing i n d i v i d u a l o p i n i o n s a l s o remains unsolved. E f f o r t s i n t h i s d i r e c t i o n have been made by Roberts (1965) and Winkler (1 9 7 1 ) , i n t e r a l i a . However, no wholly s a t i s f a c t o r y s o l u t i o n to t h i s problem i s yet i n s i g h t . In f a c t , the i s s u e would seem to have been made more complicated i n the l i g h t of the r e s u l t s of S e c t i o n 3.3 which r e v e a l that the weights could, vary with 6, the parameter of i n t e r e s t . The only f i r m recommendation which can c u r r e n t l y be made to d e c i s i o n makers seeking to f i n d a consensus fo r a panel of experts would seem to be that they should use s e n s i t i v i t y a n a l y s i s to i d e n t i f y the c r u c i a l a spects of the weight a l l o c a t i o n t a s k . 138 In a recent t e c h n i c a l r e p o r t , Zidek (1982) introduced two new c r i t e r i a f o r a s s e s s i n g group d e c i s i o n procedures. One i s based on the idea of subsampling the group and i t i s found that amongst the proposed s o l u t i o n concepts o n l y Nash's s o l u t i o n i s optimal under "subsampling." The other assumes that the group i s i t s e l f a sample from a s u p e r p o p u l a t i o n , and t h i s y i e l d s an analogue of Wald's theory where the e l i c i t a t i o n of the p r i o r s becomes part of the experimental p r o c e s s . It w i l l be i n t e r e s t i n g to see what these ideas w i l l y i e l d when they are a p p l i e d i n the present context. F i n a l l y , we l i s t a number of u n s e t t l e d t e c h n i c a l i s s u e s which were r a i s e d i n the course of the d i s c u s s i o n : 1. We proved i n S e c t i o n 2.3 that i f T i s any dogma p r e s e r v i n g s e m i - l o c a l p o o l i n g operator whose u n d e r l y i n g f u n c t i o n n "G" on [0,») i s continuous, then T i s a l s o l o c a l and hence a l i n e a r o p i n i o n p o o l . As noted i n the second paragraph f o l l o w i n g P r o p o s i t i o n 2.3.6, t h i s c o n d i t i o n on G seems ra t h e r a r t i f i c i a l and, c o n s i d e r i n g the way i n which i t was used in our proof, c o u l d c o n c e i v a b l y be weakened, i f not removed a l t o g e t h e r . How? 2. In Theorem 2.4.6, i t was seen that the l o g a r i t h m i c o p i n i o n pool i s the only a v a i l a b l e E x t e r n a l l y Bayesian q u a s i -l o c a l p o o l i n g operator when (0,M) c o n t a i n s n o n - n e g l i g i b l e s e t s of a r b i t r a r y small measure (Assumption 2.4.5). Can t h i s 139 somewhat i r r i t a t i n g r e g u l a r i t y c o n d i t i o n be weakened or, b e t t e r s t i l l , e r a d i c a t e d ? I t would be p a r t i c u l a r l y important to f i n d out whether there are or are not any other q u a s i - l o c a l E x t e r n a l l y Bayesian procedures than the l o g a r i t h m i c pool n ( a l l o w i n g f o r negative weights w as long as L w = 1 ) when 0 i i = l i i s f i n i t e and u i s a counting-type measure. 3. The problem of determining which g e A 0 maximizes the product n a 1 -a w ( i ) P = n [/f g du] i = 1 i arose i n S e c t i o n 2.5 when we were t r y i n g to determine which u~ n d e n s i t y g opt i m i z e d the expected Renyi Information I w I (f ,g) i=1 i a i of order a, a c ( 0 , 1 ) . T h i s would seem to be a hard problem, but i t may n e v e r t h e l e s s be p o s s i b l e to solve i t a n a l y t i c a l l y . 4. I t was suggested i n Chapter 3 that the r u l e (3.2.8) could be viewed as an analogue of Bayes' formula f o r updating P-f u n c t i o n s . However, arguments i n favour of i t s use c o u l d be best developed w i t h i n the framework of an axiomatic theory of p r o p e n s i t y f u n c t i o n s . These q u e s t i o n s remain f o r f u t u r e c o n s i d e r a t i o n . 140 REFERENCES ACZEL, Janos (1948) Un probleme de M.L. F e j e r sur l a c o n s t r u c t i o n de L e i b n i z . B u l l . S c i . Math.,72,39-45 ACZEL, Janos (1966) Le c t u r e s on f u n c t i o n a l equations and t h e i r a p p l i c a t i o n s . New York: Academic Press ACZEL, Janos & DAROCZY, Zo l t a n (1975) On measures of i n f o r m a t i o n and t h e i r c h a r a c t e r i z a t i o n s . New York: Academic Press BACHARACH, Michael (1973) Bayesian d i a l o g u e s . Unpublished manuscript - C h r i s t Church, Oxford BACHARACH, Michael (1975) Group d e c i s i o n s i n the face of d i f f e r e n c e s of o p i n i o n . Manaq. Sci.,22,182 -191 BAHADUR, Raghu Raj (1967) Rates of convergence of estimates and t e s t s t a t i s t i c s . Ann. Math. Statist.,38,303-324 BERGER, Roger L. (1981) A necessary and s u f f i c i e n t c o n d i t i o n f o r reaching a consensus using DeGroot's method. J . Amer. S t a t i s t . Assoc.,76,415-418 BERNARDO, Jose M. (1976) The use of i n f o r m a t i o n i n the design and a n a l y s i s of s c i e n t i f i c  exper imentat i o n . Ph. D. T h e s i s , U n i v e r s i t y of London BERNARDO, Jose M. (1979) Expected i n f o r m a t i o n as expected u t i l i t y . Ann. Statist.,7,686-690 141 BHATTACHARYYA, A. (1943) On a measure of divergence between two s t a t i s t i c a l p o p u l a t i o n s d e f i n e d by t h e i r p r o b a b i l i t y d i s t r i b u t i o n s . B u l l . C a l . Math. Soc.,35,99-109 BICKEL, Peter J . & DORSUM, K j e l l A. (1977) Mathematical s t a t i s t i c s ; b a s i c ideas and s e l e c t e d t o p i c s . San F r a n c i s c o : Holden-Day BIRNBAUM, A l l a n (1954) Combining independent t e s t s of s i g n i f i c a n c e . J . Amer. S t a t i s t . Assoc.,49,559-574 BLACKWELL, David H. & GIRSHICK, Meyer A. (1954) Theory of games and s t a t i s t i c a l d e c i s i o n s . New York: John Wiley DALKEY, Norman C. (1975) Towards a theory of group e s t i m a t i o n . The Delphi method: techniques and a p p l i c a t i o n s . (H. A. Li n s t o n e & M. T u r o f f , e d i t o r s ) Reading, Mass: Addison-Wesley DAWID, A. P h i l i p (1982) The w e l l - c a l i b r a t e d Bayesian (with d i s c u s s i o n ) . J . Amer. S t a t i s t Assoc.,77,605-613 DAWID, A.P., STONE, M. & ZIDEK, J.V. (1973) M a r g i n a l i z a t i o n paradoxes i n bayesian and s t r u c t u r a l i n f e r e n c e (with d i s c u s s i o n ) . J . R. S t a t i s t . Soc. B,35,189-233 de FINETTI, Bruno (1937) La p r e v i s i o n : ses l o i s l o g i q u e s , ses sources s u b j e c t i v e s . Ann. I n s t . Henri Poincare,7,1-68 DeGROOT, M o r r i s H. (1973) Doing what comes n a t u r a l l y : i n t e r p r e t i n g a t a i l area as a p o s t e r i o r p r o b a b i l i t y or as a l i k e l i h o o d r a t i o . J . Amer. S t a t i s t . Assoc.,68,966-969 142 DeGROOT, M o r r i s H. (1974) Reaching a consensus. J . Amer. S t a t i s t . Assoc.,69,118-121 EISENBERG, Edmund & GALE, David (1959) Consensus of s u b j e c t i v e p r o b a b i l i t i e s : The pari-mutuel method. Ann. Math. Statist.,30,165-168 FADDEEV, D m i t r i i K. (1956) On the concept of entropy of a f i n i t e p r o b a b i l i s t i c scheme ( i n R u s s i a n ) . Uspehi Mat. Nauk,11 (1),227-231 FELLNER, W i l l i a m (1965) P r o b a b i l i t y and prof i t . Homewood, 111: Irwin FINE, Terrence L. (1973) T h e o r i e s of p r o b a b i l i t y . New York: Academic Press FISHER, S i r Ronald A. (1921) On the mathematical foundations of t h e o r e t i c a l s t a t i s t i c s . P h i l o s . Trans. Roy. Soc. London A,222,309-368 FISHER, S i r Ronald A. (1932) S t a t i s t i c a l methods f o r r e s e a r c h workers (4th e d i t i o n ) . London: O l i v e r & Boyd FORTE, Bruno (1975) Why Shannon's entropy. Symp. Math. V o l . XV,137-152 New York: Academic Press FRASER, Donald A.S. (1966) S t r u c t u r a l p r o b a b i l i t y and a g e n e r a l i z a t i o n . Biometrika,53,1-9 FRENCH, Simon (1981) Consensus of o p i n i o n . Europ. J . Oper. Res.,7,332-340 143 FRENCH, Simon (1982) On the a x i o m a t i s a t i o n of s u b j e c t i v e p r o b a b i l i t i e s . Theory and Decision,14,19-33 GENEST, C h r i s t i a n (1982) E x t e r n a l b a y e s i a n i t y ; An i m p o s s i b i l i t y theorem. Tech. Rep. 82-8 I n s t i t . A p pl. Math. S t a t i s t . , UBC GENEST, C , WEERAHANDI, S. & ZIDEK, J.V. (1982) The l o q a r i t h m i c o p i n i o n p o o l . Tech. Rep. 82-7 I n s t i t . A p pl. Math. S t a t i s t . , UBC GOOD, I r v i n g John (1950) P r o b a b i l i t y and the weighing of evidence. New York: G r i f f i n , London & Hafner GOOD, I r v i n g John (1955) On the weighted combination of s i g n i f i c a n c e t e s t s . J . Roy. S t a t i s t . Soc.,17,264-265 HARDY, G.H., LITTLEWOOD, J.E. & POLYA, G. (1934) I n e q u a l i t i e s . Cambridge: Cambridge U n i v e r s i t y Press HARSANYI, John C. (1955) C a r d i n a l w e l f a r e , i n d i v i d u a l i s t i c e t h i c s and i n t e r p e r s o n a l comparisons of u t i l i t y . J o u r n a l of P o l i t i c a l Economy,63,309-321 HELLINGER, E r n s t (1909) Neue Begriindung der Th e o r i e quadrat i scher Formen von unendlichen Verander1ichen. J . Reine Angew. Math.,136,210-271 HOGARTH, Robin M. (1975) C o g n i t i v e processes and the assessment of s u b j e c t i v e p r o b a b i l i t y d i s t r i b u t i o n s (with d i s c u s s i o n ) . J . Amer. S t a t i s t . Assoc.,70,271-294 144 HOGARTH, Robert M. (1977) Methods f o r aggregating o p i n i o n s . D e c i s i o n Making and Change i n Human A f f a i r s . (H. Jungermann & G. deZeeuw, e d i t o r s ! Dordrecht, H o l l a n d : D. R e i d e l P u b l i s h i n g Co. JEFFREYS, S i r Harold (1967) Theory of p r o b a b i 1 i t y (3rd r e v i s e d e d i t i o n ) . Oxford U n i v e r s i t y Press KAKUTANI, Shizuo (1948) On e q u i v a l e n c e of i n f i n i t e product measures. Ann. Math.,49,214-224 KOOPMAN, Bernard 0. (1940) The bases of p r o b a b i l i t y . B u l l . Amer. Math. Soc.,46,763-774 KULLBACK, Solomon (1968) Information theory and s t a t i s t i c s . New York: Dover P u b l i c a t i o n s KULLBACK, Solomon & LEIBLER, R i c h a r d A. ( 1 951 ) On i n f o r m a t i o n and s u f f i c i e n c y . Ann. Math. Statist.,22,79-86 LINDLEY, Dennis V. (1956) On a measure of the in f o r m a t i o n provided by an experiment. Ann. Math. Statist.,27,986- 1005 LINDLEY, Dennis V. (1961) The use of p r i o r p r o b a b i 1 i t y d i s t r i b u t i o n s i n s t a t i st i c a l  i n f e r e n c e and dec i s i ons. Proceedings of the Fourth Berkeley Symposium on Mathematical S t a t i s t i c s and P r o b a b i l i t y , vol.I,453-466 B e r k e l e y : U n i v e r s i t y of C a l i f o r n i a Press LINDLEY, Dennis V. (1982) S c o r i n g r u l e s and the i n e v i t a b i l i t y of p r o b a b i l i t y (with d i s c u s s i o n ) . I n t . S t a t i s t . Review,50,1-26 145 LINDLEY, D.V., TVERSKY, A. & BROWN, R.V. (1979) On the r e c o n c i l i a t i o n of p r o b a b i l i t y assessments (with d i s c u s s i o n ) . J . R. S t a t i s t . Assoc A,142,146-180 LIPTAK, Tamas (1958) On the combination of independent t e s t s . Magyar Tudomanyos Akademia Matematikai K u t a t 6 Intezetenek  Kozlemenyei,3,171-197 LITTELL, Ramon C. & FOLKS, J . Leroy (1971) Asymptotic o p t i m a l i t y of F i s h e r ' s method of combining independent t e s t s . J . Amer. S t a t i s t . Assoc.,66,802-806 LITTELL, Ramon C. & FOLKS, J . Leroy (1973) Asymptotic o p t i m a l i t y of F i s h e r ' s method of combining independent t e s t s I I . J . Amer. S t a t i s t . Assoc.,68,193-194 LUCE, R. Duncan & RAIFFA, Howard (1957) Games and d e c i s i o n s . New York: John Wiley MADANSKY, A l b e r t (1964) E x t e r n a l l y bayesian groups. RAND Memo RM-4141-PR MADANSKY, A l b e r t (1978) E x t e r n a l l y bayesian groups. Unpublished manuscript - U n i v e r s i t y of Chicago MARSHALL, A l b e r t W. & OLKIN, Ingram (1979) I n e q u a l i t i e s : theory of m a j o r i z a t i o n and i t s a p p l i c a t i o n s . Nev York: Academic Press MATUSITA, Kameo (1951) On the theory of s t a t i s t i c a l d e c i s i o n f u n c t i o n s . Ann. I n s t . Math. S t a t i s t . , 3 , 1 7 - 3 5 146 McCONWAY, Kevin J . (1981) M a r g i n a l i z a t i o n and l i n e a r o p i n i o n p o o l s . J . Amer. S t a t i s t . Assoc.,76,410-414 MONTI, Kath e r i n e L. & SEN, Pranab K. (1976) The l o c a l l y optimal combination of independent t e s t s t a t i s t i c s . J . Amer. S t a t i s t . Assoc.,71,903-911 MORRIS, Peter A. (1974) D e c i s i o n a n a l y s i s expert use. Manaq. Sci.,20,1233-1241 MORRIS, Peter A. (1977) Combining expert judgments: a bayesian approach. Manaq. Sci.,23,679-693 MOSTELLER, F r e d e r i c k & WALLACE, David L. (1964) Inference and di s p u t e d a u t o r s h i p : the f e d e r a l i s t . Reading, Mass.: Addison-Wesley MUDHOLKAR, Govind S. & GEORGE, E. Olusegun (1979) The l o g i t s t a t i s t i c f o r combining p r o b a b i l i t i e s - an overview. O p t i m i z i n g methods in s t a t i s t i c s . (J.S. R u s t a g i , e d i t o r ) New York: Academic Press NASH, John F. J r . (1950) The b a r g a i n i n g problem. Econometrica,18,155-162 NORVIG, Tor s t e n (1967) Consensus of s u b j e c t i v e p r o b a b i l i t i e s : a convergence theorem. Ann. Math. Statist.,38,221-225 NOVICK, Melvin R. & HALL, W. Jackson (1965) A bayesian i n d i f f e r e n c e procedure. J . Amer. S t a t i s t . Assoc.,60,1104-1117 PRESS, James S. (1978) Q u a l i t a t i v e c o n t r o l l e d feedback f o r forming group judgements and making d e c i s i o n s . J . Amer. S t a t i s t . Assoc.,73,526-535 147 RAIFFA, Howard (1968) D e c i s i o n a n a l y s i s : i n t r o d u c t o r y l e c t u r e s on c h o i c e s under  u n c e r t a i n t y . Reading, Mass: Addison-Wesley RENYI, A l f r e d (1961) , On measures of entropy and i n f o r m a t i o n . Proceedings of the Fourth Berkeley Symposium on Mathematical S t a t i s t i c s and P r o b a b i l i t y , v o l . 1,453-468 Be r k e l e y : U n i v e r s i t y of C a l i f o r n i a Press ROBERTS, Harry V. (1965) P r o b a b i l i s t i c p r e d i c t i o n . J . Amer. Stat i s t . Assoc.,60,50-62 ROY, S.N. (1953) On a h e u r i s t i c method of t e s t c o n s t r u c t i o n and i t s use i n m u l t i v a r i a t e a n a l y s i s . Ann. Math. Statist.,24,220-238 ROYDEN, Halsey L . (1968) Real a n a l y s i s . New York: Macmillan RUDIN, Walter (1974) Real and complex a n a l y s i s . New York: McGraw-Hill RUDIN, Walter (1976) Pr inc i p l e s of mathemat i c a l a n a l y s i s (3rd e d i t i o n ) . New York: McGraw-Hill SAMUELSON, Paul Anthony (1947) Foundations of economic a n a l y s i s . Harvard U n i v e r s i t y Press SANDERS, F r e d e r i c k (1963) On s u b j e c t i v e p r o b a b i l i t y f o r e c a s t i n g . J o u r n a l of A p p l i e d Meteorology, 2,191-201 148 SAVAGE, Leonard J . (1971) E l i c i t a t i o n of p e r s o n a l p r o b a b i l i t i e s and e x p e c t a t i o n s . J . Amer. S t a t i s t . Assoc.,66,783-801 SCHOLZ, F r i t z W. (1981) Combining independent P-values. Tech. Rep. 11 Department of S t a t i s t i c s , U n i v e r s i t y of Washington ( S e a t t l e ) SEN, Amartya K. (1970) C o l l e c t i v e c h o i c e and soc i a l w e l f a r e . San F r a n c i s c o : Holden-Day SHAFER, Glenn (1976) A mathematical theory of evidence. P r i n c e t o n , N.J.: P r i n c e t o n U n i v e r s i t y Press SHANNON, Claude E. (1948) A mathematical theory of communication. B e l l System T e c h n i c a l J o u r n a l , 27,379-423 and 623-656 SHAPIRO, Alan R. (1977) The e v a l u a t i o n of c l i n i c a l p r e d i c t i o n s : a method and i n i t i a l a p p l i c a t i o n s . New E n g l . J . Med.,296,1509-1514 SHAPIRO, A.R., DANNENBERG, A.L. & FRIES, J.F. (1979) Enhancement of c l i n i c a l p r e d i c t i v e a b i l i t y by computer c o n s u l t a t i o n . Meth. Inform. Med.,18,10-14 SION, Maurice (1968) I n t r o d u c t i o n to the methods of r e a l a n a l y s i s . Toronto: H o l t , Rinehart & Winston SLOVIC, P., FISCHHOFF B. & LICHTENSTEIN, S. (1977) B e h a v i o r a l d e c i s i o n theory. Ann. Rev. Psychol.,28,1-39 STAEL VON HOLSTEIN, C a r l - A x e l S. (1970) Assessment and e v a l u a t i o n of s u b j e c t i v e p r o b a b i l i t y d i s t r i b u t i o n s . Stockholm: Economics Research I n s t i t u t e at the Stockholm School of Economics 1 49 STEIN, C h a r l e s M. (1965) Approximations of improper p r i o r measures by p r i o r p r o b a b i l i t y  measures. B e r n o u l l i , Bayes, Laplace A n n i v e r s a r y Volume ( J . Neyman & L. LeCam, e d i t o r s ) New York: S p r i n g e r - V e r l a g STONE, Mervyn (1961) The l i n e a r o p i n i o n p o o l . Ann. Math. Statist.,32,1339-1342 STONE, Mervyn (1976) Strong i n c o n s i s t e n c y from uniform p r i o r s (with d i s c u s s i o n ) . J . Amer. S t a t i s t . Assoc.,71,114-125 TVERSKY, Amos & KAHNEMAN, D a n i e l (1974) Judgment under u n c e r t a i n t y : h e u r i s t i c s and b i a s e s . Science,185,1124-1131 WAGNER, C a r l (1982) A l l o c a t i o n , Lehrer models, and the consensus of p r o b a b i l i t i e s . Theory and decision,14,207-220 WEERAHANDI, Samaradasa & ZIDEK, James V. (1978) P o o l i n g p r i o r d i s t r i b u t i o n s . Tech. Rep. 78-34 I n s t i t . A p p l . Math. S t a t i s t . , UBC WEERAHANDI, Samaradasa & ZIDEK, James V. (198 l ) M u l t i - b a y e s i a n s t a t i s t i c a l d e c i s i o n theory. J . R. S t a t i s t . Soc. A,144,85-93 WINKLER, Robert L. (1967) The assessment of p r i o r d i s t r i b u t i o n s in bayesian a n a l y s i s . J . Amer. S t a t i s t . Assoc.,62,77 6-800 WINKLER, Robert L. (1968) The consensus of s u b j e c t i v e p r o b a b i l i t y d i s t r i b u t i o n s . Manag. Sci.,15,B61-B75 WINKLER, Robert L. (1971) P r o b a b i l i s t i c p r e d i c t i o n : some experimental r e s u l t s . J . Amer. S t a t i s t . Assoc.,66,675-685 ZELLNER, A r n o l d (1977) Maximal data i n f o r m a t i o n p r i o r d i s t r i b u t i o n s . New Developments i n the A p p l i c a t i o n of Bayesian Analys (A. Aykac and C. Brumat, e d i t o r s ) Amsterdam: North-Holland ZIDEK, James V i c t o r (1982) Aspects of m u l t i b a y e s i a n theory. Tech. Rep. 82-11 I n s t i t . Appl. Math. S t a t i s t . , UBC 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            data-media="{[{embed.selectedMedia}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0080161/manifest

Comment

Related Items