TOWARDS A CONSENSUS OF OPINION by CHRISTIAN GENEST B . S p . S c , 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 M o n t r e a l , 1978 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department We a c c e p t this of Mathematics t h e s i s as to the r e q u i r e d THE standards UNIVERSITY OF BRITISH January © Christian conforming COLUMBIA 1983 Genest, 1983 In presenting requirements thesis in partial fulfilment I available for agree that the Library shall reference and study. I for extensive c o p y i n g of t h i s make further thesis her representatives. p u b l i c a t i o n of t h i s thesis It for is understood financial 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 Mathematics The U n i v e r s i t y o f B r i t i s h 2075 W e s b r o o k P l a c e V a n c o u v e r , Canada V6T 1W5 Date: January 25, 1983 Columbia gain the of British it freely agree for p u r p o s e s may be g r a n t e d by t h e Head of my D e p a r t m e n t or of f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y Columbia, permission this that that scholarly or by copying shall not his or be iii Abstract This density t h e s i s a d d r e s s e s t h e p r o b l e m of functions, f f combining , of n i n d i v i d u a l s . the prior In t h e f i r s t of n two parts, various characterize n I w f , i=1 i i = systems of axioms s u c c e s s i v e l y the l i n e a r and the logarithmic a(i) n a(i) f /JUt dy. i=1 i i=1 i It i s first n n pooling T(f 1 f operator, ...,f n is T H(O,...,O,0) in opinion shown f ), n i t s first leads n to of E x t e r n a l with c h a r a c t e r i z a t i o n of G as the o n l y T(f,,...,f f result, the f )du n for some i t i s necessary to introduce underlying space of e v e n t s , and on f f) = n the only as satisfies 6. This i n v o l v i n g Madansky's of t h i s i n some d e t a i l and axiom yield E x t e r n a l l y Bayesian pooling ){6) H:(0,») ...,f ) n expressible depend = n JH ( 1 r The r e g u l a r i t y c o n d i t i o n not a r e a l s o examined form ( is Other consequences a the G A is an i m p o s s i b i l i t y theorem group r a t i o n a l i t y of that variables of operator pool, A(f which f o r some f u n c t i o n H w h i c h i f H does Bayesianity. developed pool, which = 0 f o r n- a l m o s t a l l 6. on H may be d i s p e n s e d axiom f opinion )(6) = H ( f , ( 6 ) , . . . , f (6),e) n continuous result ( are n H(f,(6),...,f( 6 ) ) / n —>(0,=°). To a "richness" (0,M). Next, prove this condition each o p i n i o n on f is i iv regarded for as containing a pooling maximum. which The is entails operator chosen; the for The notion pool, which m o t i v a t e adapted to derive a encompasses b o t h A and G. given via A is a definition i t i s argued P-values. class final of that beyond d e n s i t i e s . and examples a r e these include A theorem pooling the of A c z e l i s formulas characterization of which G is t h e i n t e r p r e t a t i o n o f b e t t i n g odds, and t h e p a r a l l e l between our a p p r o a c h and problem" extend generalization; large look A. should well-known problem of combining content Kullback-Leibler's function i s introduced this 0 and we depends on t h e d e f i n i t i o n of the d i s s e r t a t i o n , operators of p r o p e n s i t y about information obtained example, second p a r t domain o f p o o l i n g given so opinion the "information" whose e x p e c t e d operator the l i n e a r In some Nash's solution to the "bargaining i s discussed. James V. Thesis Zidek supervisor V Table of Contents Abstract List i i i of f i g u r e s Technical vi note v i i Acknowledgements CHAPTER 1.1 viii I . PROLEGOMENA Introduction 1 1.2 The p r o b l e m o f t h e p a n e l 1.3 P r e v i o u s 1.4 O u t l i n e CHAPTER of experts proposals 3 6 of subsequent chapters 14 I I . POOLING DENSITIES 2.1 F u n d a m e n t a l s and n o t a t i o n 17 2.2 McConway's work 20 i n review 2.3 A c h a r a c t e r i z a t i o n o f t h e l i n e a r opinion 2.4 S e e k i n g pool via locality 29 E x t e r n a l l y Bayesian 2.5 I n f o r m a t i o n minimizing m a x i m i z i n g and pooling procedures divergence operators 66 81 2.6 D i s c u s s i o n CHAPTER I I I . POOLING PROPENSITIES 3.1 M o t i v a t i o n 3.2 A class 3.3 D e r i v i n g CHAPTER 47 of l o c a l 89 pooling the l o g a r i t h m i c operators opinion pool IV. SUGGESTIONS FOR FURTHER RESEARCH REFERENCES 99 124 136 140 List of Figures Two o p i n i o n s with a d i f f e r e n t entropy but g i v i n g the same p r o b a b i l i t y to the t r u e value of the q u a n t i t y of i n t e r e s t before i t i s r e v e a l e d to be one vii Technical Note T h i s t h e s i s was p r e p a r e d on t h e Amdahl 470 V8 computer of the University of B r i t i s h Columbia w i t h t h e a i d o f t h e FMT t e x t processing language. Because the c h a r a c t e r s e t s f o r the Xerox 9700 printer a r e somewhat l i m i t e d , i t was n e c e s s a r y t o d e p a r t s l i g h t l y from some conventional mathematical symbolism. For instance, t h e l e t t e r "R" had t o be r e s e r v e d t o d e n o t e t h e r e a l l i n e and t h e B r i t i s h Pound S t e r l i n g symbol " £ " was substituted for script e l l . On some o c c a s i o n s , i t was a l s o n e c e s s a r y t o w r i t e s u b s c r i p t s on t h e same l i n e as the indexed quantities, e.g. w ( i ) i n s t e a d of w when this quantity a p p e a r e d a s an i exponent. F u r t h e r m o r e , 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 t h e r e a d e r w i l l n o t be i n c o n v e n i e n c e d by t h e s e d e p a r t u r e s from common u s a g e . The m a t e r i a l i s d i v i d e d i n t o 4 c h a p t e r s , and e a c h chapter into several sections. E q u a t i o n s , d e f i n i t i o n s , t h e o r e m s and e x a m p l e s are numbered i n the decimal n o t a t i o n . T h u s , E q u a t i o n (2.3.5) r e f e r s t o t h e f i f t h l a b e l l e d e q u a t i o n o f S e c t i o n 3, i n Chapter 2. Within Section 2.3, i t might be r e f e r r e d t o s i m p l y as Equation ( 5 ) . viii Acknowledgements The a u t h o r w i s h e s t o thank h i s s u p e r v i s o r , P r o f . James V. Zidek, for suggesting most of t h e p r o b l e m s t r e a t e d i n t h i s t h e s i s and for f i n a n c i a l assistance. His a p p r e c i a t i o n i s a l s o extended to Mr. B r u c e J . Sharpe for h i s constant w i l l i n g n e s s to enter into discussion. The f i n a n c i a l s u p p o r t o f t h e N a t i o n a l Science and Engineering Research Council and of 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 a r e 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 w i t h a l l my l o v e t o t h e two women who have supported me t h r o u g h o u t my s t u d i e s : my m o t h e r , L u c i e L a p o i n t e - G e n e s t , and my w i f e , C h r i s t i n e S i m a r d - G e n e s t . 1 I. 1 .1 Introduction In this devising t h e s i s , we s h a l l methods an uncertain "opinion polls" Usually, probability "states event, (e.g., as will distributions 1966) o r f i d u c i a l provided that 1937; this to of and an the the belief" as an has recently been © of structural Indeed, individual are manner, t h e y c a n be t o t h e axioms of p r o b a b i l i t y theory question space f o r instance. i n a coherent outcome (subjective) posterior, of we for?".). appropriate be p r i o r , of i n t e n t i o n , as i n to vote expressed distributions, the "degrees quantitatively be over They m i g h t (Fraser shown t o c o n f o r m opposed problem By " o p i n i o n , " vis-a-vis "whom do y o u i n t e n d opinions the opinions. of a person's b e l i e f of n a t u r e . " assessed be c o n c e r n e d w i t h f o r aggregating mean t h e e x p r e s s i o n of PROLEGOMENA reexamined (de F i n e t t i by Lindley 1982) . However, (Winkler 1967; T v e r s k y to confirm of the that the r e l a t i v e nature, common & Kahneman and experimental likelihood generally of studies 1974; S l o v i c e t a l . 1977) t e n d a l t h o u g h an i n d i v i d u a l may have a good i t cannot calculus observation of the various be e x p e c t e d probabilities possible that and he w i l l express knowledge states also his of master opinion 2 accordingly. such widely uniform spread or concept M o r e o v e r , we- would l i k e non-informative of b e l i e f Therefore, f :0 — > [ 0 , = ° ) the range o f propensity belongs we will In expert 0. any willing opinion with who an Shafer opinion to A asked the In our be we and any will recent function = {g:0 call —>[0,»)|fgdM If it Furthermore, an to a if f =1} of measure v likelihood will them. will be on i s deemed t o be As should of all l o n g as whose enlightening, a l w a y s be be but assumed and facts i s the case, and every t r e a t e d with c o n s i d e r a t i o n . to probability an event simply what t h e a s s e s s o r b e l i e v e s i t t o be; thing as or review of "objective" c f . Fine relevant this or concerning o p i n i o n , the e x p e r t s are capable attributed approach, assessor some k i n d of e x p e r t personalistic "correct" call to produce h i s o p i n i o n assessment or we discussion, i t will for their in s u b j e c t i v e this vague, (1976). to (0,<»), literature, i s asked i t s v a l u e and a of t h e more respect to a dominating assessor so. known relative as w e l l as use 0 of p o s s i b l e s t a t e s of n a t u r e . subject matter be has improper, or P - f u n c t i o n f o r s h o r t . with to present evidence as f o r the that f i s a ^-density. the when an collection person on take restricted a say belief priors, space densities need not For, the f is Generally, that shall accordance opinion it we on to of f u n c t i o n d i s c u s s e d by function, probability 0, expressions to account (1973). the or h y p o t h e s i s there opinion. theory, is For a no is such critical 3 1.2 The p r o b l e m of t h e In group a decision of opinion panel analysis, individuals' w h i c h may judgements. be discussion bargaining Let very us little need suppose, this can distribution (1961), Zellner and a formal which adds has been work as result presented or collection decision who do this of to along Novick & not importance irreversible use well decide be and of data t o c o n s u l t one knowledge b e l i e v e d t o be the these using sample lines (1965) and r e l e v a n t to and paradoxes (1973) or Stone i f the analysis will is pressing impractical, or by non-informative I f time is the solve by satisfactory its decision. of To Hall inconsistency approach w i l l has distribution h i s ignorance Dawid, S t o n e & Z i d e k of a l a r g e amount maker may have by i s of m a j o r in a costly, group maker who analysis. done uncritical such p r o b l e m a t hand for the panel a prior little (1967), However, t h e Moreover, rules i s confronted with determine to express m e a s u r e s sometimes l e a d s t o those peoples' without p r o b l e m of Bayesian prior (1976). the choose Jeffreys (1977). decision for instance, that a decision undertake much these opinions knowledge of a s u b j e c t - m a t t e r information; Lindley single representative determining is called a (1968). p r o b l e m , he m i g h t prior a individual to quantify h i s b e l i e f s b e f o r e he a Raiffa into t o combine of as a c o n s e n s u s of of aggregating after i t i s often necessary thought statistically experts, experts beliefs T h i s problem nor of several the the people question, 4 i.e. experts. These consultants. their experts Even a f t e r beliefs and to available information, converge to say that needed has take end, from t h e collected, total that the As them about experts only of perceived the decision panel respective When t h i s in dissensus. being a their jointly agreement. does form of a l l the unlikely number of without to modification i t is how said d i s c u s s i o n amongst account group i s l e f t i n the it into a s t a t e of the extract extensive proper opinions are will happens, we opinion is one maker and proceed to (possibly) diverging opinions irresponsive to any he particular assessment? Despite making little prevalence situations, this a t t e n t i o n i n the papers of Winkler instructive raised the by 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 problem literature, (1968) introduction and to group-assessments decision for each and maker argument of which a consensus intend to propose interpretations of various of attempt to should explore such by vague i n the theoretical Raiffa will review An questions (1968). assume t h a t the of experts. have. the out place at find comparatively (1975,1977). a number, n, panel opinion and the with the i n h i s s h o e s and pointed Hogarth have t a k e n ourselves received phenomenon, we i s presented member as i s provided In c o n c e p t u a l i z i n g t h e discussion has all t i m e when opinions, We desirable will one place properties More s p e c i f i c a l l y , consequences concepts as the of we various "adequacy," 5 " r e p r e s e n t a t i v e n e s s " and The we a p p r o a c h we prescribe expert opinions c o h e r e n c e and person out view that take here way rationality. a decision to adhere No Moreover, we i n the problem would i n the p r e s e n c e of u n c e r t a i n t y about to the d e c i s i o n f ( f of how a be o b s e r v e d t o implicitly is T that postulates i s made t o d e s c r i b e are maker sense maker s h o u l d p r o c e s s to c e r t a i n attempt w i t h the a c t u a l the t a s k . interest i s normative, i n which i f he w i s h e s confronted carry of the "consensus." adopting ©, t h e ), t h e the quantity consensus n opinion the describing possible views f has Bernardo 1979 in a different clear that from t h a t of where decision and where utility and in beliefs i s by no means g e n e r a l l y some s u p p o r t e r s ( W i n k l e r 1968; situation, relations of o u t c o m e s o f t h e e v e n t upon c o l l e c t i n g attitude perfectly markedly assessment concerning the experts* f . n This it his final the -by bargaining between a s u r v e y paper group main lies invoked. 1978; makes differs decision-making w i t h the f i n a l group considerations of distinctions and The have been ... w.ell & Zidek it t o examine a necessity- problems by W e e r a h a n d i with although & Zidek However, propose faced of are t h e s e two we concern way Weerahandi context). the problem a accepted, (1981). emphasized 6 Admittedly, something open of a misnomer. the possibility "synthetic has failed the case issue a joint is of a set-up group to r e a l i z e , has to a lot amongst here, is also an opinion" against formula Previous that the but literature the panel of e x p e r t s the one hand a r e those the whose members which is called of This beliefs would be asked to in a t h a t the common with theory played by from observations and that when he quantity opinion, role are there which to judge the of attempts to number of a opinions, is no performance d e c i s i o n maker c h o o s e s t o the the "true of the use. proposals The probability then, decision estimate Only sought objective leaves d e c i s i o n maker r e p r e s e n t s produce a s i n g l e assessment standard strategies estimate are of present forecast. observations. 1.3 the the various opinions voiced. important statistician pooling that to of t h e d e c i s i o n maker devise fact, " d e c i s i o n maker" i s i f , f o r example, a g r o u p of m e t e o r o l o g i s t s was It our expression to c r e a t e a consensus, nevertheless, representative the In personality" discussion upon, therefore, which relates d i v i d e s roughly those t o the into so-called two main papers which d e a l d i r e c t l y distributions and which are concerned their with aggregation; a broader problem streams: with on picture, of on personal the other t h a t of a 7 group d e c i s i o n utility the p r o b l e m , and r a t h e r than latter helpful type on will which focus on the c o n s e n s u s of o p i n i o n s . o n l y be Stone (1961), g r o u p a s s e s s m e n t s by distributions. most taking a Formally, if f of i n c l u d e some hand. investigators weighted of References m e n t i o n e d when t h e y comments r e g a r d i n g t h e p r o b l e m a t Following considerations have represented of individual average represents the probability i density of quantity defined the 6, i - t h member of then the the linear group c o n c e r n i n g opinion pool for the n unknown experts is by n T (f where w w , > 0 and i insure f ) n n I w =1. i =1 i that the = Z w f , i =1 i i The joint (1.3.1) restrictions o p i n i o n of t h e on the group, T weights, ( f f ), i will n be a density. Winkler weights hoc for solution (Stael Von weighting (1968) d i s c u s s e s t h e problem each e x p e r t ' s d i s t r i b u t i o n schemes. In H o l s t e i n 1970; scheme, w = 1/n, subsequent Winkler was and determining proposes studies, 1971) never of found he the various ad and others t h a t the uniform outperformed more than i marginally (in terms of predictive ability) by other schemes 8 which attempted or p a s t for performance combining supporting that and Formula composite t h a n most be t o rank e a c h e x p e r t a c c o r d i n g classical i t was to ordering differences essentially irrelevant 1 experts, 1 observed e m p i r i c a l fact the a of a v e r a g e p o i n t sensible possible of o p i n i o n ability might e s t i m a t e s i n the and (1.3.1). which the group w i l l alternatives. he is After solid opinion pool relation action the in as a u n i q u e which he an a c t i o n affected arguing on solution, by A that by the in favour of this attributes (1.3.1) and for face finds conditions i t s members d o e s , not linear that preference derives prefer 1975) i s mainly concerned with entail, In a t h e o r e m Bacharach relation of utility, (1973, the group courses relation (1964), that predictive a phenomenon w h i c h Although Bacharach B whenever e a c h o f preference Also greater f o r implementing emerge. preference Madansky show Bayes' (1965) estimation. grounds of expertise Theorem). i n t h e work of B a c h a r a c h Stone's Formula action i s the o f t e n distributions t h e o r y of the e x i s t e n c e group (1.3.1) his t h e method o f R o b e r t s updating weights using to the r e l i a b i l i t y theoretical began included of t h e i n d i v i d u a l linked But (these to to assuming to another the group's presence of of these D r . F. P. G l i c k b r o u g h t t o my a t t e n t i o n some recent work of Alan S h a p i r o (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 h i m s e l f and u s e d t h e l i n e a r o p i n i o n p o o l t o i n c r e a s e d i a g n o s t i c a c c u r a c y of p h y s i c i a n - e x p e r t s . 9 postulates, he goes on t o show t h a t 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 i n t r o d u c e d , i n t o d i c t a t o r i a l form, i . e . one the p o o l can of the w 's of be forced (1.3.1) e q u a l s 1 i while the remainder are 0. t h a t the group a c t s as i f maximizer. In This far-reaching condition it by our endeavour, we have been very much s t i m u l a t e d by the Madansky (1964,1978) E x t e r n a l In a framework we propose the to will similar prove concept Bayesianity. work of K e v i n McConway (1981) who j u s t i f i c a t i o n for using too expected-utility called using we single an theorem" 2.4, a such "impossibility Section were asserts was the f i r s t t o g i v e a linear adopt opinion pool ourselves. strong within the H i s main theorem s t a t e s t h a t i f a d e c i s i o n maker wants h i s p r o c e s s of consensus f i n d i n g t o commute w i t h 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 t o use Formula Chapter thesis 2 of this will start with (1.3.1). a d i s c u s s i o n of McConway's r e s u l t . Despite is endowed i t s great p o p u l a r i t y , the weighted average w i t h f e a t u r e s which may viewed as drawbacks. (1.3.1) is For in c e r t a i n circumstances instance, Winkler notes that t y p i c a l l y m u l t i - m o d a l on i t s domain and so may fail l e a d s him c a l l s the be (1968) t o 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 This formula 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 n a t u r a l c o n j u g a t e approach. group member's o p i n i o n choices. In this method, each i s deemed t o c o n s t i t u t e "sample e v i d e n c e " 10 which can Doksum 1977, process be manner The 77) interest. similar to will for each of opinions are It distribution order on is precisely t h a t even multi-modal with the pushes W i n k l e r ' s probability revealed these distribution, a one with complex that bases decision and maker. knowledge and own the are using members good of which that To treats weights proceed his Morris is entirely each expert the rely the assessors on is the consensus to introspect opinions of the one-by- A conventional, In panel sources. assessment their rule. their whose v a l u e obtain then us In h i s work, M o r r i s results. probability leaves brought use and combine Bayes' analysis data a obviates it random v a r i a b l e representing uni-Bayesian where a function his and philosophy. maker must experts' approach the degree t o which step further as in Theorem. determining of e x p e r t the d e c i s i o n likelihood different i d e a one the of difficulties Bayesian opinions of B a y e s ' this question & the group assessment, overlapping experience distribution to if (Bickel data-generating distributions, (1974,1977) t o e l a b o r a t e a t h e o r y consistent the t o form e x p e r t s , as w e l l as based prior maker need o n l y combine the d i f f i c u l t the of successive applications of with a n a t u r a l conjugate In recognize possibility nonetheless by to the the d e c i s i o n reader the p. of therefore, represented simplest independent ( i . e . they albeit case, data are 11 calibrated experts' ), M o r r i s 1 priors shows t h a t 1 f ...,f ) a(i) = opinion proposed an on ad pool hoc f i=1 1,1£i£n; t h i s logarithmic a( i ) n = n with normalized product of the obtains: n T(f the n /; i a( i ) n f is a particular which basis. (1.3.2) d*x, i= 1 i Dalkey Note case of (1975) that the s o - c a l l e d had for T(f earlier f ) t o be a n M-density in (1.3.2), However, M o r r i s members of the the solution constitute an panel, which would be acceptable consistent and represents a practical assessment problems this ways of It 1 other the do not need t o add d e c i s i o n maker forces a(l) = expert-dependent ... is up one = a(n); and to thus one. of the otherwise, could not consensus. work c e r t a i n l y insightful, reason, pooling a(i)'s assumes t h a t Although Morris' For the only few methodology is conceptually would a g r e e w i t h because which o v e r l a p p i n g researchers of the experience have c o n t i n u e d appealing, him that i t insurmountable would to seek cause. direct opinions. is interesting to observe that the p r e s c r i p t i o n embodied As L i n d l e y & a l . (1979) o b s e r v e d , t h i s i s a s k i n g a l o t . On t h e other hand, Dawid (1982) has r e c e n t l y shown t h a t a c o h e r e n t B a y e s i a n e x p e c t s t o be w e l l c a l i b r a t e d ! ! 1 2 in but (1.3.2) i s n o t o n l y as w e l l This the commutativity updating axiom, of by Madansky the into (1978) and approach survive group of using Bayesianity External probability; a still another whilst i t will opinion procedure pool The b a s i c that formulas of those p o o l i n g literature, of only the dictatorships, natural conjugate contains opinions is by logarithmic argue that i n some a r e so d i s c r e p a n t that s a t i s f a c t o r y means f o r c h o o s i n g 2.5, a s i m i l a r i d e a Section some s e n s e , the p r i o r pool because, will 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 f o r combining opinion they t o be i n t r o d u c e d of t h e e x p e r t s be shown i n Their an i n c o r r e c t ( b u t c o r r e c t a b l e , Moreover, the only In S e c t i o n i s , in Weerahandi i t p r i o r - t o - p o s t e r i o r coherence. Bayesianity. coin" them. and ( i i ) formula. h a s a l s o been s u p p o r t e d r a n d o m i z e d d e c i s i o n r u l e s need "tossing compounding explored 2.4.6) d e r i v a t i o n o f t h e the p r i o r (i) requires the t e s t . manuscript Theorem the a d d up t o p r o c e d u r e s have been applications ( 1 9 7 8 ) , who c a l l unpublished finds, in certain External & Zidek who discussed (1.3.2) between a analysis (1964,1978), operations of E x t e r n a l l y Bayesian Madansky cases, when t h e a ( i ) ' s p r o b a b i l i t y assessments v i a Bayes' which a r e cf. formulated probabilities properties by i m p l i c a t i o n of M o r r i s ' i t i s E x t e r n a l l y Bayesian one. individual a natural 2.4 that the only density lead opinion the to pool, logarithmic Externally functions us f Bayesian f of n n individuals. 13 Our survey densities of the two 1959, the on complete Eisenberg scheme the on the without appropriately situations. & Gale problem at l e a s t of pooling a brief for (see a l s o N o r v i g combining "pari-mutuel" betting principles determining determine However, opinions w i l l and be literature mention f o l l o w i n g papers. ingenious based the would not In an of allow their for that reason, be probability method. T h e i r i d e a was judgements shown that in that odds" could more general certain holders to d i c t a t e the p a r i - m u t u e l presented distributions "totalisator group i t can 1967) individual the consensus method has never odds, been very popular. Then, D e G r o o t the d i s t r i b u t i o n s updates his importance then are weights what using further and to determine i t i s when under which the theme extensive iterative i s d e s c r i b e d by list consensus i t does e x i s t . Berger p a p e r and process Press of r e f e r e n c e s . The (1978), any of is the theory exists p o i n t s out an exact c o n d i t i o n s converge. who expert procedure distribution (1981) of assigning Markov C h a i n g i v e s the will by longer a l t e r t h e o r e m s from when a i n DeGroot's o r i g i n a l no apprised p a n e l , each (1.3.1) h i s peers. revisions limit error the Formula t o h i m s e l f and until t h a t upon b e i n g t h e o t h e r members of t h e prior opinions, invoked and of own iterated members' (1974) p r o p o s e s also A variation provides on an 14 Although D e G r o o t ' s method does not (remember that imagined that decision consulted. important linear (ii) flaws it is External (iii) As 0. about no of of the ratings that to French general that the no value as a of subsequent out enlarge the we concerned functions. chapters. the beliefs, The the each the various individual who were ( 1 9 8 1 ) , however, we find three approach. They are: p r o p o s e d as an hoc outside 8 by people ad data, (i) the procedure; observations or is available (this invalidates leaves the first and last but (non-negligible) group s e c t i o n of t o what c o u l d p r o b l e m of out be in not case dissensus. chapters i n the of be other selection principle); procedure ourselves we the p r o v i s i o n i s made f o r t h e expression density the panel t o have been e x p r e s s e d Rather, will w e l l as and of set-up i t could carried taken note is s t i l l assumed pointed restrict assumed pool iterative Outline form in t h i s is has with Bayesianity when t h e not as process once he to himself opinion least, iterative Along information 1.4 the expressed granted f i t our assume a l l d i s c u s s i o n i s c l o s e d ) , maker h i m s e l f , opinions expert we formally as of be probability of f:0 — > [ 0 , » ) , with propensity two topics will chapter, called experts, definition this the distributions to i n t e g r a b l e or functions discussed do classical where o p i n i o n s opinion be we over include not. as in are any Thus, well as separate 15 Starting begins with of the short led with proof opinion of to in Section founded study Bayesianity and on Section of we of to yet "locality." Section Section 2.6 2.4 is i s shown t o be pooling of the which general procedures regard an formula are derived information contained and the Kullback's logarithmic (1968) n o t i o n discusses these 3 addresses the possessed by a in a set of operator formulas In opinions. then opinion containing limiting of d i v e r g e n c e . f i n d i n g s and way. statistic Stone's l i n e a r as the this as pools one with the maximizing p o o l i n g same External on a consensus d i s t r i b u t i o n information of is consistent properties then devoted opinion pool A least logarithmic linear using At the the a n o t h e r c h a r a c t e r i z a t i o n of derived are d e r i v a t i o n of F u r t h e r m o r e , a l a r g e c l a s s of p o o l i n g the We axiom Bayesian 2.5, idea pool. of concept 2.2. (1964;1978) w h i c h c o n d e n s e s the both the a different Property. Madansky's Some Externally leads 2.3 Marginalization in Section 2 characterization of quasi-local postulate. The v i a the d e n s i t i e s , Chapter (1981) for generalization i s explored. a only pool h i s theorem a p p e a r s to propose avenue p r o b l e m of p o o l i n g a d i s c u s s i o n o f McConway's linear operator the cases is Finally, r e i t e r a t e s some words caution. Chapter number of Section propensity 3.1 appears the which more g e n e r a l functions. motivate well-known problem Examples this of p r o b l e m of are combining presented g e n e r a l i z a t i o n ; amongst combining independent a in them t e s t s of 1 6 hypothesis. where we local. of A l l these focus our attention H e r e , we a r g u e Hardy, local scales for and to comparability & P6lya suited are assumption Section operators propensities; to the 3.2, which a r e weighted (1934) a r e t h e o n l y means "sensible" approach is t h o s e c a s e s where t h e e x p e r t s ' intercomparable. characterizing a p p r o a c h and Nash's on p o o l i n g in the q u a s i - a r i t h m e t i c aggregating well of b e l i e f devoted that Littlewood rules axiomatic also examples a r e c o n t i n u e d the The logarithmic i s n o t met; t h e last section pool parallel is when the between our (1950) s o l u t i o n t o t h e b a r g a i n i n g problem i s discussed. Finally, research. Chapter 4 contains suggestions for further 17 II. 2.1 F u n d a m e n t a l s and Throughout (mutually of n this assessors dominating (n>2) function measures, t h e r e complete and all ©} c to loss densities on 0, we will each and a of a ffdn = probability \i that 1. process in of group is opinion i s expressed for be n of to be continuous with for sets. _ so a generating assuming o-finite, a as Because M measurable of d e r i v a t ive (Sion for which each assume with a-field essential space {Ac © | M ( B ) = M ( B n A)+M(B\A) = usually 0 the to produce of g e n e r a l i t y the write A over asked that is absolutely i s not will be Q(M) taking will d e n o t e by Caratheodory Radon-Nikodym finiteness We © and the be u v which a will f:0 —>[0,°°) i s no in applications, have of will of n a t u r e For convenience, properties measure c h a p t e r , we measure on M-measurable B notation exclusive) states distribution. the POOLING DENSITIES that In every other r e s p e c t t o ii will 1968, p. of a l l 110). But o~ otherwise. the collection (f,,...,f ) ^-measurable to represent e i t h e r a typical n n element f(0) = of A n or e l s e (f , ( 0 ) , . . . , f the (0)). function The f:0 —>[0,°°) interpretation defined will by always be n clear from convention. the context, It will so no also be confusion assumed will that arise A * 0, from this so t h a t there 18 A e J2(M) exists with 0 < y(A)< » . n By A a pooling operator on 0, we mean any a p p l i c a t i o n T:A w h i c h maps t h e n - t u p l e (f f ) to T(f f ), n density. pooling The following operators which Definition 2.1.1 We a pooling (1) say t h a t operator most lists often n T:A — > A the p r o p e r t i e s of refer to. is local iff there [0,®) e x i s t s a Lebesgue-measurable such that T(f 1 f ...,f n represents the usual G:[0,=°) function ) = Gn(f,,...,f n (2) u~ a n definition we w i l l —> ) y-a.e. n —> Here, n composition of f u n c t i o n s . quasi-local iff there exists a function Ttf,,...^ )«C(f,,...,f n (3) unanimity iff T(f 1 f n C:A — > ( 0 , = ° ) such that ) i s local, n preserving ...,f ) = f y - a . e . whenever n f =f y - a . e . i for a l l 1<i<n. (4) dogma iff preserving S u p p ( T ( f f general n )) c u S u p p ( f ) n i =1 i Supp(f) = ( 0 e 0 | f ( 0 ) * 0}. y-a.e., where in 19 (5) a d i c t a t o r s h i p iff there exists T(f,,...,f ) = f n the M-a.e. that f o r a l l c h o i c e s of f , , . . . , f i in n domain o f T. To prove elementary Their 1 ^ i ^ n such theorems, results we w i l l from the p r o o f s a r e t o be f o u n d often make u s e o f t h e Theory i n Aczel of f u n c t i o n a l following equations. (1966). Lemma 2.1.2 n Let of h:R — > R be L e b e s g u e - m e a s u r a b l e i t s n variables. a l l x and constants c c domain, x = (x in each If h(x)+h(y) for or non-decreasing y, = h(x+y) vectors e R such n ...,x ). 1 f of r e a l that (2.1.1) numbers, there exist n l e x over i t s i=1 i i a l s o h o l d s t r u e when t h e h(x) The r e s u l t then = n n domain o f h i s [0,°°) or [0,K] w i t h K > 0 a c o n s t a n t . Lemma 2.1.3 n Let each h:(0,») — > R be L e b e s g u e - m e a s u r a b l e of i t s n v a r i a b l e s . or non-decreasing If h ( x ) - h ( y ) = h(x-y) for in n c(i ) a l l x,y > o, t h e n h ( x ) = II x i =1 i (2.1.2) f o r some c ( i ) e R. 20 Equations Cauchy's and (2.1.1) and f u n c t i o n a l equation. m u l t i p l i c a t i o n of v e c t o r s (2.1.2) a r e u s u a l l y In t h e p r e s e n t i s taken r e f e r r e d t o as context, addition t o be c o m p o n e n t w i s e . 2.2 McConway's work i n r e v i e w In this opinion s e c t i o n , McConway's (1981) d e r i v a t i o n of t h e l i n e a r pool (labelled will 2.2.4) w i l l m o t i v a t e and s e r v e It be d i s c u s s e d . also be offered. as a background has a l r e a d y A short proof The of h i s theorem discussion will f o r o u r own r e s u l t s . been m e n t i o n e d that t h e a p p r o a c h a d o p t e d by McConway fits the d e s c r i p t i o n of the problem of the panel of experts set out 1. To j u s t i f y the prescription could be condition be embodied called or marginalization pooled If then dogma p r e s e r v i n g introduces what (MP). This t h e same c o n s e n s u s d i s t r i b u t i o n s h o u l d d i s t r i b u t i o n s are f i r s t marginalization ( i i ) each assessor the condition first Postulate ( i ) the a s s e s s o r s ' i n t o a consensus a mild of Chapter Marginalization some and 2 i n ( 1 . 3 . 1 ) , McConway s t i p u l a t e s that and consensus; Section the a r r i v e d a t whether combined a in is performed individually resulting on t h e performs the marginal d i s t r i b u t i o n s are distribution. tantamount pooling operator to that is i n our d e f i n i t i o n of added, then McConway 21 proves he that calls turn the Strong holds form the 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 true (1.3.1). Function property This fact However, t h e p r o o f the last A h a s oik) has is dominating in terms of However, usually i s of the introduces read of his paper. The t h i r d "any S e l w h i c h l i n e of contains and n o t "any S e l c o n t a i n s A, McConway d o e s n o t assume t h e e x i s t e n c e o f measures does say t h a t a g r e e on some o b v i o u s little not i s lost entirely Stone (1961) o r W e e r a h a n d i that for some natural exist, new, process be c o n t a i n e d a-field in will fi(y). over to are stated densities. the experts will 0," and we c l a i m t h a t itself just i t s existence. as well, T h i s p o i n t of a s an i n s p e c t i o n o f t h e s e t - u p i n & Zidek (1978) w i l l function then opposed impose i t suffices a-additive his results " i npractice, by a s s u m i n g is to as a natural y will view will 3.1 measure y, and c o n s e q u e n t l y McConway Caratheodory in his Marginalization by a m i s p r i n t . p . 411, s h o u l d probability y which sub-a-algebra..."). most o f t h e t i m e , so t h a t McConway to c o n s t i t u t e s Theorem In h i s t r e a t m e n t , a (SSFP), (WSFP) and p r o c e e d s t o show equivalent as a s u b - o - a l g e b r a . . . " o(A) as a program, Property" i s obscured paragraph, Property i f the pooling operator (To c a r r y o u t h i s this Postulate. Function i f and o n l y "Weak S e t w i s e that Setwise t o what confirm. f o r the e x p e r t s T on a ring a u t o m a t i c a l l y extend Cf. Sion 1968, p . 6 7 ) . (Note t o a g r e e on H. The T t o y and H 22 d o m i n a t i n g measure u on When a n a t u r a l SSFP c o n d i t i o n Definition can 2.2.1 be, f o r m u l a t e d (McConway as 0 e x i s t s , McConway's follows: 1981) n A pooling operator T:A —>A has the Strong Setwise Function n Property such (SSFP) i f f t h e r e for all e 0(M), A indicator function Before own: we function F:[0,1] — > [ 0 , 1 ] We a three that the set in general 1(A) stands for the A. space (0,M) i s tangible y-measurable neighbourhoods A 0 < (2.2.1) n where of d/i] fI(A)f make a d e f i n i t i o n of 1 2.2.2 say = F [ f I ( A ) f ,d/ti, s t a t e McConway's Theorem, we Definition (I) a that / I ( A ) - T ( f , , . . . , f )du n our exists ) < M(A i =i 1 f i f f there A ,A 2 3 exist i n 0 such (at least) that ,2,3; i 1 S h o r t l y a f t e r t h e c o m p l e t i o n of t h i s t h e s i s , a paper of Carl Wagner (1982) was b r o u g h t t o our a t t e n t i o n i n w h i c h t h e a u t h o r u s e s the t e r m " t e r t i a r y s p a c e " t o d e n o t e what we c a l l a t a n g i b l e space. Theorem 7 of h i s p a p e r i s e q u i v a l e n t t o t h e formulation of McConway's Theorem p r e s e n t e d i n Theorem 2.2.4 below. 23 and (II) n A uik i might less following Lemma The " i n t a n g i b l e " space to t h i s question looks like. i s provided by A the 2.2.3 Proof: Let = A * 0 {0,}, the and class (®,v) y-measurable be s e t s A,,A 0 of = 2 {9, ,t5 ) v with 2 intangible with 2 properties the = counting spaces. i n t a n g i b l e , so t h a t t h e r e i s enough t o g u a r a n t e e are ( I ) and existence a t most (II). of a t least two Assuming one such A,. there clearly exists one 1 1 equivalent the other properties that only f = I(A )/^i(A ) function On wonder what an lemma. measure t y p i f y If i=j. c o m p l e t e answer s p a c e s 0, set 0 => > j One more or ) any to i s the satisfying (I) & o n l y element of ( I I ) , then A and the (Q,a) is (©,,»>). hand, ( I ) and s e t A, if there ( I I ) , then y-measurable subset are e x a c t l y two i t f o l l o w s from B of A must obey s e t s A,,A 2 Definition y(B) with 2.2.2 = 0 or n(B) i = n(k i ), i = 1 ,2,3, measure 0 or In A 3 = 0\(A,UA ). Moreover, A 2 itself 3 has ». particular, M(V(X) n where A ) = M(A if ) feA and V(x) = f o r a unique value {t9e0|f(0) of x £ 0, = x}, say x then , unless 24 y(A i ) = 0. C o n s e q u e n t l y , any + x *I(A ), 2 (0,M) i s equivalent i.e. 2 Evidently, judge interest reader can (0,M) i s i n t a n g i b l e and ( I I ) , we With precise to (0 ,i>). and practical from Examples 2.5.5 shall includes say t h a t these d e f i n i t i o n s , statement of we and two x «I(A ) 1 1 They a r e , importance, 3.1.3, s a y . If a with 2 as the space properties dichotomous. a r e now McConway's sparse. s e t s A,,A i t is as • 2 i n t a n g i b l e spaces are rather however, of some ( I ) and f i n A c a n be w r i t t e n in a p o s i t i o n to give result (the proof space. A pooling a appears below): Theorem 2.2.4 Let (0,y) (McConway 1981) be a t a n g i b l e m e a s u r a b l e operator n T:A —>A has t h e SSFP i f f n T(f f ) = n for some w £ 0 , i n I w i =1 i i=1 because spaces if 0 (0,M) had the is only could above, transparent. is be two marginalization this i i =1. McConway o b s e r v e s t h a t tangible M-a.e. Z w f restriction hardly to relevant what to neighbourhoods, performed. That In Theorem call h i s argument no view we nontrivial o f Lemma 2.2.4 cannot 2.2.3 be 25 generalized t o i n t a n g i b l e spaces is illustrated by t h e f o l l o w i n g Example 2.2.5 Suppose 0 = {8 6 } yt Let and u = c o u n t i n g 2 G:[0,1] —>[0,1] be s u c h 2 measure. that 0 i f x<y; G(x,y) = x i f x=y; 1 i f x>y, and consider T:A —>A d e f i n e d by T ( f , , f ) ( 0 ) 2 Then T has t h e Strong to check that T(f,,f ) = w,f, + w f 2 Proof o f Theorem One implication A i , A 2 and consider , A be 3 Setwise Function fora l l f , f 2 n 2 1 Property t h e r e a r e no w e i g h t s w, and w 2 = G(f (0),f (0)). 2 2 2 and i t i s e a s y i n [0,1] f o r which ' A. 2.2.4: is three obvious. To prove neighbourhoods with the other one, l e t p r o p e r t i e s ( I ) and ( I I ) , 3 f = Z a (j).I(A ) j=1 i j i where (O'/xfA,) a = x i y >0 i f o r given (2)«/u(A ) , a i x ,y i i 2 = y i in [0,1] with (2.2.1) i m p l i e s J*I ( A , ) 'T( f f )dM = F ( x ) , n (3)«y(A ) and a i 3 i x +y <1, 1<i<n. i i = 1-x - i Then 26 ;i(A ).T(£ ...,f 2 )dju = F ( y ) , 1 r n and also )du /I(A,UA )-T(f,,...,f 2 = F(x+y), n where x=(x,,...,x ), y = ( y . . . , y Therefore Using F ) and 1 f n x+y=(x,+y x +y ). n n n satisfies Cauchy's Lemma 2.1.2, i t f o l l o w s functional that equation F(x,,...,x ) = n (2.1.1). n I w x i=1 i i on n [0,1] f o r some w,,...,w n each of i n R. i t s components, w As F must be non-decreasing > 0 f o r a l l i=1,...,n; moreover, in the i fact that T(f f ) i s always normalized forces n At guards first sight, seems s e n s i b l e . against inconsistency performing a accomplished either same r e s u l t two typical i s o b t a i n e d by and It i s a principle probability The b e f o r e or a f t e r situations arise, of "marginal a n a l y s i s . " the will MP n I w = i =1 i assessments route. i n w h i c h t h e need • which when n e c e s s a r y p o o l i n g can the m a r g i n a l s are either 1. be reported; However, t h e r e a r e for a marginal they both c a s t doubt on the v a l i d i t y i s a product parameter space over which each analysis of t h e MP principle: Case I: 0 a group o f e x p e r t s has distribution. been a s k e d t o produce However, t h e d e c i s i o n maker his i s only member o f "multivariate" interested in 27 a particular variable. In that appears c o u n t e r - i n t u i t i v e , at in differing It could fields be, specialized for represents member the of panel least i n the only has one suggested h i s maker feel compelled to reduce "events"? The t o him that case, answer (and certain a l t e r n a t i v e s w h i c h had can be now r u l e d out impossibilities" (Koopman new he information, Theorem, the only distributions, Given updating seem to rational in general p r i o r s using be called the Bayes' for, assessor each decision 0 ( M ) , the of in the of new effect that possible a Upon b e i n g his c l a s s of light panel) to the apprised beliefs priori logical of using this Bayes' probability 1 processes r u l e are here, has and the p r e s c r i p t i o n for updating not. experts knowledge. have become " a s s e r t e d update s u b j e c t i v e or that is: 1940). would would been c o n s i d e r e d because they MP "univariate" distribution size to the one are situation when the prior that interest. assessment In revealed assessors only v a r i a b l e of d e c i s i o n maker. evidence i f the that to the possible McConway e x p l a i n s hence have d i s p a r a t e instance, knowledge Case I I ; 8 all 1 and case, is not not of marginalizing equivalent, what and would so much McConway's MP In this regard, c f . French (1982) who provides axioms justifying the use of Bayes' Theorem when "changes of information take the form of t h e o c c u r e n c e o f an e v e n t i n t h e f i e l d upon w h i c h t h e s u b j e c t i s c o n c e n t r a t i n g . " 28 c o n d i t i o n as an axiom which would guarantee that the same consensus distribution is p r i o r s are combined f i r s t a r r i v e d at whether ( i ) the and final experts' the r e s u l t i n g consensus o p i n i o n i s updated; or e l s e ( i i ) the p o s t e r i o r s are d e r i v e d by each individually and already e x i s t s , Bayesianity. then pooled and For by the d e c i s i o n maker. Madansky (1964,1978) calls T h i s axiom it External a more t e c h n i c a l d e f i n i t i o n of t h i s concept, as w e l l as an a n a l y s i s of some of i t s consequences, i s r e f e r r e d t o S e c t i o n 2.4 Focusing expert on see reader below. (2.2.1) now, we the a c o n d i t i o n which i s e s s e n t i a l l y equivalent to MP, that the p r o b a b i l i t y consensus d i s t r i b u t i o n , T ( f ,,...,£ ), to any assigned by the measurable set A e n fl(n) i s assumed to depend s o l e l y on the p r o b a b i l i t i e s given by the i n d i v i d u a l assessors' distributions f f . to A As n is potentially condition does terms of the dictate and that the local i s a r b i t r a r y in 0 ( M ) , t h i s A what M ( A ) >> (A's with this is T ( f one. Indeed, 1) at the same t h a t i t should f the does only ) in n 1), i t a l s o c o n t r o l s i t s unnecessary and ) not behaviour of T ( f , , . . . , f 0 < M ( A ) << f 's (A's with i g l o b a l behaviour examine large i s obviously a far-reaching (2.2.1) think very on atoms of time. We suffice to n (for n definition, c f . Royden s e c t i o n , our first linear opinion 1968, task pool by will p. 321). consist assuming only Thus, i n the of following characterizing what we the have e a r l i e r 29 defined as " l o c a l i t y " amounts r o u g h l 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 t o (2.2.1) r e s t r i c t e d 2.3 A c h a r a c t e r i z a t i o n o f t h e l i n e a r The purpose of t h i s section pool dogmas. B o t h d e f i n i t i o n s were g i v e n "locality" i s the only mimics analogous property Roughly local Bernardo speaking, opinion pooling pool operator in Section (1979) of u t i l i t y t o atoms. via locality i s to e s t a b l i s h that opinion which who u s e s the l i n e a r which 2.1. preserves The term i t to describe an functions. locality reduces pooling operators to n Lebesgue-measurable a fairly and strong correct not have constitutes I t could intuitively appealing v i a b l e a l t e r n a t i v e t o McConway's principle for pooling and t h e c o n s e n s u s p r o b a b i l i t y a t 8 i s r e q u i r e d t o depend the f o l l o w i n g the p r o b a b i l i t i e s obtained measurable defines a is be v i e w e d a s a l i k e l i h o o d upon t h e p r o b a b i l i t i e s We But i t constitutes o f 8eQ i s in upon present G on [0,°°) , and t h u s sense: a p a r t i c u l a r value operators only requirement. certainly SSFP. functions but i s not material say of those d i d not. needed to that that assigned a to the and could The c o n d i t i o n t h a t G be L e b e s g u e here, of the e n s u i n g a consensus d e n s i t y state," s t a t e s of n a t u r e which explicitly pooling "true i t ties the sections. operator which but i s zero preserves on t h a t dogmas part of if i t the 30 space where t h e p r o b a b i l i t y zero. The observes word that "dogma" assessors a l l said i s borrowed i f the p r o b a b i l i t y t o be z e r o by an i n d i v i d u a l , event say, not will rules wish ever affect out i t s f o r c e t o debate, professing at dogmas i n advance, this However, i t seems r e a s o n a b l e their the advice problem t o be d e c i d e d will by would n o t arise only that We we do whether scientific that a decision experts who shall of putatively be assessed about H i s judgement, the to expect Difficulties was evidence the open of a r t i c u l a t e common dogmas. expressed with a l l questions event (1973) i . e . i t i s doqmatic. point, clashes of l e a v i n g sought no p e r t i n e n t i t should Bacharach of a c e r t a i n his opinion. attitude has from that "facts." maker who challenge i f t h e dogmas are c o n f l i c t i n g . Note t h a t a l o c a l pooling operator need not always preserve n dogmas; c o n s i d e r f o r instance the operator map to I ( 0 ) / M ( 0 ) . everything next lemma, dogma T:A — > A However, a s w i l l preservation is automatic which be shown when would i n the M(0) i s inf i n i t e . Lemma 2.3.1 n Every M(0) local pooling operator T:A — > A preserves dogmas when is infinite. Proof; Let n G:[0,°°) — > [ 0 , ° ° ) be t h e L e b e s g u e - m e a s u r a b l e function whose 31 existence such is guaranteed that by D e f i n i t i o n = K f o r some A and observe that 1 = = 1. is conclude preserving. We to we number Since If A e R ( M ) i s 0<K<°°, l e t f = I ( A ) / K e /T(f,...,f)du G(0,...,0)•M(0\A) infinite, real 2.1.1. = G(o)-n(0\A) KG(1/K,...,1/K) is finite and t h a t G ( o ) = 0, i . e . t h a t + u(Q\h) T i s dogma • will u s e methods from the theory of f u n c t i o n a l equations prove: Theorem Let (0 local > r 2.3.2 pooling We Lemma Let be t a n g i b l e . i) split operator The l i n e a r opinion which p r e s e r v e s the proof pool is the only dogmas. o f Theorem 2.3.2 into two lemmas. 2.3.3 A L F A 2 , A properties be 3 three ^-measurable ( I ) and ( I I ) . If neighbourhoods n T:A — > A preserves in 0 with dogmas and n there such exists a Lebesgue-measurable f u n c t i o n G:[0,») —>[0,°°) that T(f,,...,f ) n then G(x) = 3 Gn(f,,...,f ) y-a.e., n n n I w x f o r a l l x e [0,1/M] , i =1 i i M(A ),u(h )}. 2 = where M = min(M(A,), 32 Proof: Call = M ( A ) , i m i loss i = 1,2,3 of g e n e r a l i t y and s u p p o s e t h a t M = m . There i s 3 i n assuming A,,A ,A 2 disjoint: 3 i * j => A A = n i 0. no j Consider f 3 L a (j)-I(A ) j=1 i j = i where a (j) £ 0 3 Z a (j)«m j=1 i j and i p r e s e r v e s dogmas, G ( o ) = 0 and /T(f,,...,f )dM 1 f o r a l l 1<i<n. n As T furthermore 3 I G(a(j))-m j=1 j = where a ( j ) = ( a , ( j ) , . . . , a = (j)), = 1 (2.3.1) j = 1,2,3. n Define n h:[0,l] —>[0,1] h(c) = by 1 - M-G((1-c,)/M,...,(1-c )/M) n for It a l lo ^ c < T will suffice Equation n I w c i = 1 i i , so t h a t h ( o ) = 0 and h ( T ) = t o show t h a t (2.1.1), for f o r some w ^0, this fact, In a(3) = particular, choices a + satisfies Lemma 2.1.2 n and I w = i = 1 i note GladD-m, whenever then i To e s t a b l i s h h Cauchy's will 1 2 = imply because t h a t by E q u a t i o n G(a(2))-m 1. functional that h(c) = h(T) = 1. (2.3.1), h(c) (2.3.2) (T-c)/M. observe (1) = x /m, that and a if (2) = o £ x < c ^ T (c -x )/m 2 are g i v e n , the imply 33 Gfx/mJ.m, However, t a k i n g + G((c-x)/m ).m 2 h(c). a n d a (2) = 0 a (1) = x /m, i = 2 i in (2.3.3) (2.3.2) shows i that G(x/m,)'m, and, similarly, = h(x) a (1) = 0 choosing i establishes a (2) = (c -x i i )/m 2 i that G((c-x)/m )«m 2 Consequently, (2.3.3) = 2 h(c-x). becomes h(c) for with = h(x) + h(c-x) a l l o -l x < c < i , w h i c h we c a n r e w r i t e a s h(x+y) = h(x) + h(y) n n f o r x, y i n [ 0 , 1 ] w i t h x+y e [ 0 , 1 ] (take This concludes the proof. • y = c-x). Lemma 2.3.4 Suppose inf{ji(A)|A possible Since measurable (II). Let three properties where M = Proof; is tangible, e J M M ) and 0 < M ( A ) < » ) } . to find w h i c h have +6, (0,M) that ^-measurable 6 > 0, M let = 0 i t i s always neighbourhoods ( I ) and ( I I ) a n d a r e s u c h t h a t M A 0 1 R A 2 , A 3 < M <M 0 min{M(A,),M(A ),M(A )}. 2 3 (0,M) i s t a n g i b l e , there neighbourhoods In f a c t , Given and we m i g h t 6 > 0 be g i v e n ; A,,A ,A 2 as w e l l 3 exist with at least properties assume t h a t we d i s t i n g u i s h two c a s e s : three u~ ( I ) and they a r e d i s j o i n t . 34 Case I; M Choose look > 0 0 B e at fi(jx) B t o be s u c h t h a t = of M , and t h e r e 0 M(B ) > M , i that f o r 2M 0 B, , A 2 Case II; M Pick B e O(M) ^ M(B)< M 0 Bn A , i = 1 ,2,3. i i definition M 0 + min{6,M } 0 M ( B )=0 i Then and 0 or £ M by t h e 0 c a n be a t most one i , s a y i = 1 , > M(B)> 3 I u(B ) . i=1 i Our t h r e e such sets are and A . 3 = 0 0 S O that 0 < M ( B ) < 6, a n d , once a g a i n , let B = i B n A , i =1 ,2,3. i sets A i A ,A ; 2 3 Put m = max{u(B^ ),u(B ),M(B )} 2 i f m>0, Theorem then r e p l a c e at intangible Example Let © = above. Note first, space. also our c h o i c e of s e t s i s B, • consequence o f t h e two lemmas the t a n g i b i l i t y that, not every An example contrary pooling to this t o what operator effect of the space one might i s local on a n follows. 2.3.5 { 0 T , 0 2 and } Let T:A —>A 2 2 i s not l o c a l . 1 M = counting be d e f i n e d 2 and T( f , , f ) (c9 ) = but B by B,. 2.2.5 shows t h a t iscritical. conjecture I f m=0, 2.3.2 i s an immediate a b o v e , a n d Example (0,M) M ( B , ) = m. ,B so t h a t i and r e l a b e l the 3 ~ f i ( 0 i ) - f 2 measure, as i n by T ( f , , f ) ( 0 , ) 2 ( 0 i ) . Then Example = 2.2.5 f,(6,)•f (6,) T preserves 2 dogmas 35 It way. the i s possible to generalize In t h e p r o p o s i t i o n definition of l o c a l i t y depend on 6 a s w e l l f stated take at that below and p r o v e d i s relaxed a s on t h e v a l u e s point. 2 . 3 . 2 in at least Theorem to allow that i n the sequel, the f u n c t i o n the d e n s i t i e s T h u s , we c o n s i d e r one a l l pooling G to f operators n n T:A — > A o f t h e form ) ( e ) T(f,,...,f = G(e,f,ce),...,f n (e)) n-a.e. (2.3.4) n n for some operator measurable which G:0x[O,°>) — > [ 0 , ° ° ) . function satisfies (2.3.4) will be c a l l e d A pooling semi-local. Theorem 2 . 3 . 6 Let (6,M) pooling be t a n g i b l e , operator which and suppose preserves that T is dogmas. I f a semi-local G(0,O the of n Equation almost (2.3.4) a l l 8 e ©, t h e n T i s a l i n e a r 2.3.6 Theorem pooling i s c o n t i n u o u s as a f u n c t i o n operators says that i s small. 3 and M i s a c o u n t i n g opinion the c l a s s Indeed, measure, on [ 0 , = ° ) for pool. of s e m i - l o c a l non-local when © i s c o u n t a b l e , i t i s in fact u~ empty. |6| ^ F o r , take x n [0,1] e f i (e) = x i and let 6, 77, X be = 1 - f (X) and g (7?) = x i i i /Gn(f ,...,f )du 1 n = three elements = 1-g ( X ) , i /Gn(g ,...,g )dn 1 n = the in facts 0. If that 1 and f (X) = g ( X ) , i i 3 6 together 1 = 1 , . . . , n 2 . 3 . 6 Theorem arbitrary It G is is an = G(t9,x) attempt to Our G(i?,x). generalize this proof of argument t o spaces. i s conceivable could that entail that the extra (continuity) condition be weakened, b u t we h a v e n o t a t t e m p t e d whatever necessary n-tuple be ( 0 , M ) , to rule of o p i n i o n s the requirement that T preserves out p o o l i n g o p e r a t o r s (f t o do s o . Note dogmas w h i c h would map f ) t o t h e same f i x e d on every y-density g e n A. These operators correspond up to a r e worse the case than dictatorships, where t h e d e c i s i o n maker's mind i n a d v a n c e and " c o n s u l t a t i o n " i s c o n d u c t e d only. They would Here for t h e r e f o r e seem t o be o f l i t t l e again, the since result fails to extend they i s made form's sake interest. t o dichotomous spaces: Example Let 0 = 2 . 3 . 7 { 0 , , i 9 G : 0 X [ O , 1 ] + 2 2 } , u — > [ 0 , 1 ) G(6,f,(6),f (6)) 2 measure a n d c o n s i d e r d e f i n e d by G ( 0 , , x , y ) ( 1 -x ) • I {y | 0<y< 1 / 2 } (1-x)-I{y|0<y<l/2}. it = counting and Then T : A G(t9 ,x,y) 2 2 — > A a function = x•I{y|0<y<1/2 = x-I{y|y=0 defined by i s s e m i - l o c a l and p r e s e r v e s or T ( f 1 1/2<y< 1 } , f dogmas. o r y=l} 2 ) ( 0 ) However, i s not l o c a l . The gist of the proof o f Theorem 2 . 3 . 6 i s contained i n + 37 Lemma Let 2.3.8 A,,A ,A 2 preserves function ^-measurable neighbourhoods n ( I ) and ( I I ) , and l e t T:A — > A properties which be t h r e e 3 such that be a p o o l i n g n I f G:0x[O,°°) — > [ 0 , ° 0 dogmas. (2.3.4) holds in for a l l choices 0 with operator i s a measurable of f f in n A, t h e n there exist w,,...,w n G(«,x) n I w x i=1 i i = n )] a l l x e [0,1/M(A for e [0,1] and n Z w =1 i =1 i with M a.e. - on A f o r which (2.3.5) j j=1,2,3. j Proof: We d i v i d e the proof into three parts. n Step 1; Define we w i l l show (2.1.1). F i r s t note f(x) that that f if f i ;T(f,,...,f satisfies = x I(A,) i Cauchy's functional + y I(A ) i + z I(A ) i 2 the fact g = y i h I ( A y i s i n A, )dy = / I ( A , )G( • ,x)di* + J l ( A ) G ( • , y ) d y that T preserves )+ 2 /I(A )G(•,z)dM 3 dogmas. 2 3 =1 Letting M(A )]•I(A )/u(h ) [1-y i = 3 Equation 2 n + by for a l l x e [0,1/M(A,)] ; = / I ( A , )G( • , x)d/i and 3 i [ M ( A 2 ) / * I ( A 1 ) ] . I ( A 1 ) •+ [1-y M ( A 2 ) ] . I ( A 3 ) / M ( A 3 ) (2.3.6) 38 both i n A,1^i<n, /T(g,,...,g )&n we see t h a t = n ;i(A )G(-,y)dM 2 and + ;i(A )G(. [T-y/*(A )]/M(A ))dM 3 2 f = 1 3 also )dM /T(h,,...,h = n /I )d/i + ( A , ) G ( • ,yu(A )/u(h,) 2 ;i(A )G(-,[T-yM(A )]/M(A ))dM 3 2 = 3 1. Thus JI(A )G(•,y)dM = 2 = f(yM(A )/M(A, JI(A )G(' yM(A )/M(A ))dii f 1 Similarly, we find ( A ) G ( •,z)d*i /I 2 1 3 now ( 2 . 3 . 6 ) f(x) whenever + X M ( A , ) Relabelling y ( 2 . 3 . 7 ) that /I(A,)G(•,zn(h )/n(h,))d so ) ) . = 3 and 2 = f(ZM(A )/M(A,)) ( 2 . 3 . 8 ) 3 reads f(y (A )/ u(A M + M 2 Y M ( A j 2 ) 1 ) ) f( Z M ( A + + z/u(A ) = 3 = yn(h )/M(A,), z 2 = 3 ) / M ( A 1 ) ) = 1 1. zn(A )/n(A,), we 3 have f(x) + n f(y) + f(z) = 1 f o r a l l x,y,z in [0,1/M(A,)] with x + y + i i z = l / * i ( A , ) , 1<i<n. i n So i f u, v a r e i n [ 0 , 1 / M ( A , ) ] if z = (1/M(A,))-U n with u + v e -v , i = 1 , . . . , n , then f(u+v) [0,1/M(A,)] , and + f(o) + f(z) = 1 39 and also = 1. f(u) + f(v) + f(z) But a g a i n , f(o) = 0 because T p r e s e r v e s dogmas, and so f ( u + v ) = f ( u ) + f ( v ) . t o Lemma 2 . 1 . 2 now, According there e x i s t a,,...,a in R such n n L a x i =1 i i that f(x) = these constants a on [0,1/M(A,)] n , and s i n c e f ( x ) > 0 always, are non-negative. i Furthermore, w i fCT/VtA,)) n Z a i=1 i = 1 , so t h a t = Just AI(A,). put = a /M(A,),1<i<n. i 2: We show t h a t G(0,x) i s jtz almost (9 on A = A , . For that, we use the Step measurable subset everywhere constant i n - key fact that for any u~ A' of A , n /I(A')G(•,x)dM = I w M(A'). x i =1 i i If 0 M(A')= /u(A), = M(A), this or we c a n a p p l y i s obvious, t h e above argument /I(A')G(•,x)dM = and i f 0 < n ( A ' ) < t o see t h a t n L w'x M (A') i =1 i i and n /I(A")G(•,x)dM where A " = A \ A ' and t h e p r i m e s = Z w"x M(A"), i=1 i i on t h e w 's i n d i c a t e a possible i dependence on the s e t over p r i m e s may, in fact, which G(«,x) be d r o p p e d ; for, if i s integrated. These 40 g x = i [1-x I(A') + i M(A')]-I(A )/M(A ) 2 2 i and h = x i are [ (A')/M(A") ]-I (A") + [1-x ( A ' ) M i M i i n A,1<i<n, /T(g,,...,g equals JT(h 1 r which )d/x = / I ( A ' ) G ( • , x ) d M + J l ( A ) G ( • , [ 1 - X M ( A ' ) ] / y ( A ) ) d / i n 2 equals )du ;I(A")G(.,XM(A')/*i(A"))dM = we /I(A )G(.,[T-XM(A')]/ (A ))dM, 2 M 2 conclude /I(A')G(•,x)dM and 2 then + from 2 2 n 1 and ...,h (A )//i(A ) M /I(A")G(•,XM(A')/y(A"))d = M in turn n Z w'x M ( A ' ) i=1 i i for every possible choice n Z w"x i=1 i i = o f x,,...,x M(A') i n [ 0 , 1//u(A) ] . Thus n = w" i f o r a l l i=1,...,n, /I(A)G(•,x)d and m o r e o v e r w' = w i i Jl(A')G(•,x)dM + = M w' i since /I(A")G(• x)dM f entails n Z n w x M(A) = Z i = 1 i i or n Z w x i=1 i i n Z w'x i=1 i i w'x [/i(A')+M(A") ] , i = 1 i i f o r a l l x,,...,x i n [0,1/JU(A)]. n n Finally, the fix x e [0,1/M(A)] s e t A ' = {0eA|G(0,x) > and s u p p o s e t h a t n Z w x +6} i =1 i i f o r some 6 > 0, i s non-negligible. Then 41 n /I(A')G(•,x)dM = n Z w x n(A') > I i = 1 i i a contradiction. similar S t e p 3; so argument We Hence G ( « , x ) steps x n Z w x i =1 i i n~a.e. 5M(A') , on A, and a inequality. 1 and 2 f o r A or A 2 n )] a l l x e [0,1/M(A n Z w x i =1 i j i = 3 i n s t e a d of A , , ju-a.e. and some g i v e n constants j satisfying But M(A') + that G(«,x) for ^ shows t h e r e v e r s e can repeat w i = 1 i i by n Z w i=1 i j (2.3.9) w i n [0,1] ij =1, j =1,2,3. (2.3.7), n /I(A )G(•,y)dM = 2 Z w i=1 = M(A ) y 2 i2 i /I(A,)G(•,yM(A )/M(A,))dy 2 n Z w i=1 for a l l y e [0,1/M(A )] 2 y i1 i w i 1 the stated , so t h a t Let = w follows i3 conclusion. £ denote 2 n w i 1 Similarly, u(A ) from • the c o l l e c t i o n = w , i=1,...,n. i2 (2.3.8) and so (2.3.9) entails 42 {A, eJ2(/n) IA! (note that ,A 2 ,A 3 have properties £ * 0 <=> ( 0 , M ) strengthened for some A (I)&(II) i s tangible). Lemma 2 ,A 3 CJ2(M) } 2.3.8 can be i n t h e f o l l o w i n g way: Lemma 2.3.9 Let (0,M) be tangible n and l e t T:A — > A be a dogma preserving n semi-local pooling measurable function e [0,1] ,...,w n I f G:0x[O,<=°) — > [ 0 , ° ° ) operator. f o r which n Z w =1 i= 1 i satisfying G ( • , x) (2.3.4) h o l d s , n Z w x i =1 i i = denotes the then there and such exist w, that M~a.e. on A , n w h a t e v e r be x e [ 0 , 1 / M ( A ) 3 and A e £. Proof : Let A and B i n and G(«,y) sets of = follows If be so n Z w'y i=1 i i weights, n [0,1//z(A)] £ that G(«,x) y - a . e . on B each set , y i n [0,1/M(B)] adding = n Z w x i =1 i i M a.e. on A _ f o r some {w },{w'} i n [ 0 , 1 ] i i up to 1, and a l l x in n . That w = w', i the {w } i and {w } i 1 exist from Lemma 2.3.8. /i(AnB)> 0, then c l e a r l y i i = 1,...,n. Otherwise, let 43 A, be a n o n - n e g l i g i b l e that fact A 1 A ,A f 2 that of A \ B subset have p r o p e r t i e s 3 (0,/z) i s tangible. and p i c k ( I ) and ( I I ) . From Lemma A ,A 2 3 e 0(M) SO H e r e , we u s e d t h e 2.3.8 a b o v e , we know that G(-,x) n I w x i=1 i i = n )] where x e [ 0 , 1 / M ( A M - a . e . on A , j (2.3.10) a n d j =1,2,3. j If u(B n A ) j > 0 f o r j=2 o r 3, we a r e done, {B,A ,A } If not, then and ( I I ) a n d we employ Lemma 2 G(«,y) y being 3 = constitutes 2.3.8 a g a i n n Z w'y i=1 i i to conclude that j i - a . e . on A , j n )] i n [0,1/M(A arbitrary a s e t with properties (I) a n d j=2,3. (2.3.11) Pulling (2.3.10) j and (2.3.11) t o g e t h e r We say that Z c I ( A )fi-a.e. i^1 i i of disjoint Proposition If n T:A —>A on shows t h a t a /u-density w' = w , i = 1 , . . . , n . i i f e A i s a simple f o r some c £ 0 a n d a s e q u e n c e i sets. With this definition, • function { A efl( M ) | i= 1 ,2. . .} i we c a n s t a t e and p r o v e 2.3.10 i s a dogma p r e s e r v i n g a tangible space (0,ju), t h e n semi-local there pooling that n Z w = 1 and i =1 i operator 6 exist n such i f f f= [0,1] 44 T(f 1 f ...,f ) = n for all f 1 f ...,f simple n Z w f i=1 i i functions u-a.e. i n A. n Proof: If f f e A a r e simple functions, i t i s possible to find a n sequence with = { A efi(u)|i=1,2,...} i S constants O^c i=1,...,n. Since so A that the <=° for of d i s j o i n t which f = sets A j Z c I(A ) j2M i j j ij i (Q,n) i s t a n g i b l e , we c a n assume t h a t 's b e l o n g to f . together T h u s , by Lemma 2 . 3 . 9 , /n-a.e., |S| > 2 , there exist j e [0,1] summing up t o 1 f o r w h i c h ( 2 . 3 . 5 ) holds true. n Since A T i s semi-local, f o r each T(f,,...,f j £ 1, and o b s e r v e ) = G(«,c ,...,c ) M a . e . on n 1j nj - that c j that G(',c ,...,c )= 1j nj The Dr. formulation Harry Corollary If (6,M) linear n Z w c M a . e . on A by Lemma i=1 i i j j - of the f o l l o w i n g Joe (personal « M ( A ) < J f d/x ^ 1 , i j j i corollary so 2 . 3 . 9 . was s u g g e s t e d by communication). 2 . 3 . 1 1 i s tangible opinion pool and u i s b o t h is the only o-finite semi-local and atomic, pooling the 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 atoms is that at (0,M) and atomic, most c o u n t a b l e ; is tangible. So, the collection furthermore, i f we write |C| C ^ = C of its 3 from t h e fact | i = 1 ,2, 3.. .}, e v e r y w h e r e as an {A i every function infinite only. f e A can be expressed sum L c I(A ), i*1 i i i.e. A Apply Proposition 2.3.10. In 2.3.10 that any dogma p r e s e r v i n g local. What is clear, respect to and Lemma pointwise Proposition condition necessarily however, i s secured simple follow from semi-local i s that if T convergence 2.3.10 a r e by of pooling topology, equivalent. requiring G itself Proposition operator i s continuous This to be then is with Theorem regularity continuous. 2.3.12 n Let functions • i t d o e s not 2.3.6 - consists general, the ju almost T:A n —>A be semi-local corresponding function c o n t i n u o u s as a and f o r which l e t G:0x[O,°°) — > [ 0 , = ° ) (2.3.4) h o l d s . n function on [0,°°) for M almost f —>f ik Proof: Let pointwise i ix-a.e. as ( f f G(6,-) a l l 6 e 0, _ Lim T(f , .. . , f ) = T k—>°> 1k nk whenever If ) n k—>», be i = 1,...,n. the is then 46 A = B= n U i=1 {0e0| Lim k—>» {0e0|T(f U k>1 (0)#f f ik )(0) ,...,f 1k (c9) } , i * G(0,f (0),...,f nk 1k (6))}, nk n and C = { 0 e 0 | G ( 0 , « ) Let also i s n o t c o n t i n u o u s as a f u n c t i o n D = {0e0|T(f,,...,f )(0) on * G(0,f,(0),...,f by E t h e y - n e g l i g i b l e have Lim T(f ,...,f )(0) k—><=° 1k nk = }. (0))} n denote [0,°°) and n For a l l 0 s e t AUBUCUD. e 0\E we Lim G(0,f (0),...,f (0)) k—>°° 1k nk = G(0,f,(0),...,f (0)) n )(0), = T(f,,...,f n i.e. Lim T(f ,...,f ) k—>=° 1k nk convergence topology. To c o m p l e t e = ) 1 in the pointwise n • the p r o o f of combine P r o p o s i t i o n T(f ,...,f Theorem 2.3.6, it suffices to 2.3.10 w i t h t h e above lemma, k e e p i n g i n mind that every non-negative measurable limit o f some sequence of s i m p l e function functions on a s p a c e 0 (Royde.n i s the 1968, p. 224) . In 0, a dominating natural that M, and conclusion, choice case, we have a r g u e d measure n will that usually f o r b o t h t h e e x p e r t s and opinions the c o n d i t i o n when p o o l i n g we called on impose itself as a the d e c i s i o n maker. In t a k e t h e form o f d e n s i t i e s which opinions with respect "locality" (or to perhaps 47 semi-locality) axiom seems more r e a d i l y i n t e r p r e t a b l e t h a n McConway's (2.2.1). opinion pool McConway s can then be a p p e a r s as Theorem 2.3.2. so-called "semi-local" have an important Madansky's characterization 1 idea reformulated consequence of External These following Bayesianity will be linear locality extends t h i s operators. i n the the i n t e r m s of Theorem 2.3.6 pooling of and result findings section, to will where studied at some length. 2.4 Seeking E x t e r n a l l y Bayesian In seemed Section a 2.2, more than present section, concept and McConway's decision-making Bayesian perceived will Bayesianity as an axiom problems. within our Basically, procedure as to acting give a some characterizes (1964;1978) criterion will conditions External experts. we that External for Marginalization investigate particular, interpreted suggested appropriate formulas Bayesianity we procedures the (EB) be logarithmic of The group the i n the a under opinion of this consensus, In External pool (2.4.2). problem pf were t o use manner of the which by Madansky for solving be readily however, can f o r the In implications. rationality panel determine definition introduced concept, framework if been pooling Postulate. its stated has selecting precise of Bayesianity the an they panel of Externally would a s i n g l e Bayesian. be This 48 entails To updating insure necessary their that that they the whether they pool light new of beliefs i n accordance w i t h Bayes' would a c t pooling before or in a consistent procedure after yield updating fashion, the their rule. same i t is result beliefs in the T is information. More p r e c i s e l y , we Definition have t h e following 2.4.1 n Let T:A —>A Externally 0 be a pooling operator. We say that Bayesian i f f < / * T ( f f )dn < », and n T [ * f , / j * f ,dM,...,*f / J * f djz] = n n * T ( f f ) / J # T ( f f n whenever f<i>f dy i * : 0 < » likelihood — > [ 0 , = ° ) f o r each i s a ^-measurable 1<i<n M-a.e. M (2.4.1) (such a function function $ such t h a t is called 0 < a function). Examples dictatorships opinion )d n of and Externally (provided Bayesian i t i s well defined) procedures the are logarithmic pool, T(f,,...,f ) n = n w (i) n w (i) n f // n f du, i=1 i i=1 i n L w(i)=1. i=1 (2.4.2) 49 In h i s book on d e c i s i o n a n a l y s i s , what can happen probability (on own a if distributions interest consensus opinions on They do the of need n o t entirely be win." to expect so the In c a s e relevant selfish and and update opinions before the possibility their o p i n i o n s over Their might that his the who rationale the new that call for by present will example their Such by gaining this maker, the desire reasonable is right. However, of i s found both the whether to should not some a d v a n t a g e strategic axiom by coherence." from the opinion experiment, p o s t e r i o r which w i l l prior. Weerahandi his prior earlier for manoeuvring. derives i s a Bayesian, his future their behaviour welcomed -by a consensus an quite the q u e s t i o n by of decision only the outcome of "prior-to-posterior the i t in impact a l s o been a d v o c a t e d using i n v e s t i g a t i o n s as the the a l w a y s be evidence it use find i t would be experts i f each e x p e r t he an gives experiment. w e l l have been h i s p o s t e r i o r i n an similarly, He w o u l d b e l i e v e he B a y e s i a n i t y has (1978) observation the disagree, or a f t e r of pooling to maximize the d e c i s i o n maker- and admit and experts motivated should illustrates updating l e a r n s of perceived of (1968) d e c i s i o n maker t o compute he outcome experts Zidek the in order evidence the commute. before they of i n w h i c h two t h a t each expert External not consensus regardless & do to convince distribution experiment. new, processes dichotomous space) best "to the Raiffa result and from 50 Thus, a l l i n a l l , eminently reasonable operators. easy Bayesianity seems to 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" 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 be an pooling w i t h an lemma. Lemma Let External 2.4.2 n T:A — > A be an E x t e r n a l l y Bayesian pooling T p r e s e r v e s dogmas a n d f u r t h e r m o r e T{t,,...,t ) operator. = Then T(g,,...,g n M -a.e. whenever f i = g u~a.e. i ) n f o r a l l 1^i<n. Proof: Let e A be s u c h f,,...,f Z = { 6 e © | f A 6) = ... = f that n (0)=O} non-negligible (i.e. M(Z)>0). If $ = I(0\Z), then *f = i so /4>f i du - J $ T ( f f is n 1 , 1<i<n. )dM = K ( f Using for Equation some (2.4.1), real number f i t follows 0<K<« and i and that also n T f ) = $ T ( f f n But the right-hand S u p p ( T ( f f To prove side )/K n-a.e. n 0 y - a . e . on Z , so t h a t equals n )) c u S u p p ( f ), i . e . T i s dogma n i =1 i the second assertion, suppose t h a t f i let = {0e©|f A i i (0) = g (6)} i n U A and $ = 1 ( A ) . i =1 i i n S2(M), preserving. = g y-a.e. i i=1,2,...,n. Define and A = 51 uih) Since = 0, $f Consequently, fi-a.e. = f i and similarly ffcf d/n i = )dM =1, and find using the )dM hypothesis that T i s E x t e r n a l l y Bayesian, *.T(f ... f ) = ) Ai-a.e. ) = $'T(g ,,. .. ,g ) n n e v e r y w h e r e , and h e n c e u~a.e. 1 r r n also T(<t>g,, — However, ^ f = 4>g i ,<i>g i T(f 1 r ...,f ) = T(g 1 r ...,g n Section 2.2 answer. Had Theorem Let pooling • cause a panel view of that new experts to update v i a B a y e s ' Theorem t h a n McConway p o s t u l a t e d Postulate, sometimes External he information to yield Bayesianity their marginalize the same instead would have o b t a i n e d a of very result: 2.4.3 (©,M) our both procedures w i l l Marginalization different conveyed distributions them, a l t h o u g h u~a.e. ) n above more r e a l i s t i c a l l y probability his we that n will =1. n T(*f,,...,*f and jx-a.e. i furthermore = / * T ( g g n now, = g i f $ g dju i ; * T ( f f And #g i be (An I m p o s s i b i l i t y Theorem) tangible. operators The only are d i c t a t o r s h i p s . Externally Bayesian local 52 Proof: n Let T:A —>A be local from Lemma 2.4.2 that T(f,,...,f n I w «f i=1 i i ) = n a consequence of and E x t e r n a l l y Bayesian. T also preserves /i-a.e. Then we dogmas, and for some w £ hence n I w =1 i=1 i 0, i Theorem 2.3.2. We show t h a t know w = as 1 for some j j = 1 ,..., n. Let A 1 f A because Now e S2(M) have p r o p e r t i e s 2 (0,M)is pick tangible, i * j and we i n { l , . . . , n } and ( I ) and can (II). take Such s e t s them t o consider f be exist disjoint. = I(A,)/n(A,), f i I(A )/M(A ) 2 If where k r u n s o v e r 2 * = x-I (A, ) + yI(A ) 2 Equation (2.4.1) a p p l i e d w /x = i Assuming that w the set of k indices f o r some x and y = in {1,...,n}\{i}. (0,°°), x*y, then on A, i m p l i e s t h a t w /[w «x + (1-w )«y]. i i i is neither 0 nor 1, we conclude that x=y, a i contradiction. Remark 2.4.4 Under holds that • the for (6,M) pooling semi-local be of of Theorem pooling tangible operator This previous hypotheses operators. i s indispensable, Example of above Note t h a t as result the also condition evidenced by the 2.2.5. theorem g e n e r a l i z e s finding 2.3.6, t h e Genest Weerahandi & (1982) and Zidek conflicts (1978). In with a their 53 manuscript, these logarithmic authors opinion pool Bayesianity and l o c a l i t y . only when true function 1(0) call emphasize that regarded as Bacharach's Theorem the on the External 2.4.3, a r e 0 b u t one, a n d we 2.4.3 an d i c t a t o r s h i p s of desirable. this is i n t e r p r e t any characteristic "Impossibility opinions cannot function In t h i s E x t e r n a l l y Bayesian Theorem" to generally be I n d e e d , we would be i n c l i n e d (1975) p o l i c y on t h i s m a t t e r and make N e x t , we e x t e n d of both of space. Theorem inadmissible. operators derivation based In view of a l l w 's i o f t h e whole a (2.4.2) r a i s e d t o t h e power 0 a s We no proposed case, local t h e t h e o r e m would pooling our search to follow dictatorships read: "there are operators." f o r E x t e r n a l l y Bayesian t o the c l a s s of q u a s i - l o c a l p r o c e d u r e s , i.e. pooling operators t h e form T(f,,...,f ) = n Gn(f f ) / / G n ( f ! , . . . ,f )dji n n /z-a.e. (2.4.3) n where the G:[0,°°) — > [ 0 , ° ° ) rather distinctive i s a Lebesgue-measurable property that function with 54 0 < TGn( f f n for a l l choices of f 1 r ...,f )dju < » i n A. This (2.4.4) definition of quasi- n locality chapter is equivalent to that ( D e f i n i t i o n 2.1.1). multiply by non-zero p o s i t i v e constant We local have a l r e a d y pooling (2.4.2). that and Note t h a t could any top = i b o t t o m of operator, A ) n n n [*f i= 1 n = provided n Z w(i) i =1 is however, a met, smaller =1. c l a s s of ] /K i beginning i s not of unique, side this as we of (2.4.3) E x t e r n a l l y Bayesian quasi- a l t e r i n g T. logarithmic i f *:0 pooling formula —>[0,») order n /;* is such n i i du that restrict f w ( i) du ] /K w ( i) n f to ^-densities f [#f i=1 i )/J*T(f,,...,f to ensure i t i s necessary n n i=1 w ( i) i=1 i *T(f,,...,f In w ( i) /; i $ n f = the then = n G without namely t h e < =>, at right-hand n w ( i) IT x and i =1 i = f<i>f du i T(*f,/K,,...,*f the e n c o u n t e r e d one H e r e , G(x) 0 < K given )du n Condition the (2.4.4) domain of T to f o r which the integral n w ( i) J IT. f d(i i=1 i i s always n n JUt i=1 w (i) du i is strictly positive (that 55 finite follows from Holder's w(i)'s are non-negative; c f . Marshall H e r e , we have c h o s e n A both for simplicity analysis suggest {feA|f*0 and a fair at t Olkin least when 1979, p . the 457). t o use = 0 inequality, ease amount M-a.e.} of e x p o s i t i o n . about If A the behaviour 0 * 0, o u r of E x t e r n a l l y n Bayesian q u a s i - l o c a l pooling knowing from procedures to A dogmas, fails 0 some should the they experts have) somewhat maker pathological conventional functions situation i s that "impossible" but exchanged Moreover, to reconcile have d i s j o i n t analysis support fact, Bayesian which (zero the That probability) this t h e i r views occasion ( a s we have i ti s unrealistic the i r r e c o n c i l a b l e . s i t u a t i o n i s indeed Bayesian In where some e v e n t E n o t by a l l . i s unlikely. the d e c i s i o n only t o encompass a r i s e a f t e r the experts assumed expect of on A . quasi-local Externally the JHju) w o u l d have been deemed by in 2.4.2 t h a t preserve restriction in Lemma operators acting not u n l i k e that when t h e p r i o r and to This faced likelihood a n d some i m p r o v i s a t i o n i scalled for. The problem which we will now a d d r e s s i s : a r e t h e r e any n Externally besides Bayesian q u a s i - l o c a l (2.4.2)? The answer pooling operators T:A 0 —>A 0 i s no, a t l e a s t when one i s w i l l i n g 56 t o make an extra Assumption 2.4.5 There exist small about non-negligible (0,M), namely in 6 /n-measurable s e t s of a r b i t r a r y measure, i . e . V 6 e (0,») I n d e e d , we the assumption logarithmic Theorem 2.4.6 Suppose (0,ji) opinion pool 3 e 0(M) A will now such prove pooling that the 0 < M(A) following < 6. (2.4.5) characterization of operator: satisfies Assumption (2.4.2) i s the only 2.4.5. The logarithmic E x t e r n a l l y Bayesian q u a s i - l o c a l n pooling operator Remark 2.4.7 If (0,M) and © satisfies —>A . 0 0 Assumption is infinite. important is T:A 2.4.5, t h e n c l e a r l y Thus Theorem c a s e where © is finite. 2.4.6 The i t is above d o e s not answer i n the tangible cover latter the case unknown. A s p e c i a l case establishing Lemma Theorem the following lemma w i l l prove u s e f u l in 2.4.6: 2.4.8 Suppose exists of (©,M) a sequence satisfies (A Assumption efi(M)|n=1,2,...} n 2.4.5. of Given mutually 5 > 0, disjoint there sets 57 such that 0 < M(A ) n < 5 f o r a l l n £ 1. 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 disjoint /i-measurable < 6, i = 1 , . . . , n , l e t B e Sl(u) neighbourhoods with such 0 < M(A ) i that 0 < M(B) < (1/2)-min{M(A )|l<i<n} < 6. i Then M(B) < 8, M ( A \ B ) £ M ( A )/2 i i forms a c o l l e c t i o n Another function o f n+1 m u t u a l l y obvious consequence G i n (2.4.3) must and so {A,\B,...,A \ B , B ] n disjoint sets of Assumption be d e f i n e d in . • 2.4.5 i s t h a t t h e n on (0,°°) : everywhere Lemma 2.4.9 Suppose (©,M) s a t i s f i e s A s s u m p t i o n x,,...,x be g i v e n i n (0,<=°). 2.4.5, a n d l e t There e x i s t f f n that in A 0 such n n M( n {0e6|f (0)=x }) > 0. i =1 i i Proof: Write 6 = min{l/x |l<i<n} a n d u s e Lemma 2.4.8 to choose A e i n(/x) such given jx-density, that 0 < M ( A ) < 8 a n d u(Q\k) then JhI(0\A)dM > 0 > 0. If h e A f o r otherwise 0 h i s any would 58 vanish on some set of strictly positive measure, a contradiction. Def i n e f = where p = We -hi(0\A) p i i=1,...,n. Clearly start Proposition e f i A 0 i n n ( 0 e 6 | f (0)=x } = A i=1 i i and + i M(A)]/fhI(0\A)dM, [1-x i -I(A) x i the proof is non-negligible. of Theorem 2.4.6 • with 2.4.10 n Suppose (0,At) Externally satisfies Bayesian and Assumption of the form 2.4.5. I f T:A (2.4.3) f o r 0 —>A some 0 is Lebesgue- n measurable and G:(0,=>) — > ( 0 , » ) , then G(cx) = cG(x) f o r a l l c^O n e (0,°°) . x Proof: If c=0, then s u p p o s e c>0 G(o)=0 and by Lemma 2.4.2 l e t x>o be (T preserves dogmas). So fixed. -1 Given 6 = min{[x (c+1)] | l < i < n } , we can use Lemma 2.4.8 to find i five Let disjoint 7 > 0 be elements such A , B , C , D , E of 0 ( M ) w i t h measure that -1 7 < min{2[x -J»(A)-CM(B) ] / M ( A U B ) i } in (0,6). 59 and pick 0 < X,i; < » so t h a t -1 X < min{-7+2^[x ~u(A)-cu(B) ] / u ( A U B ) | 1 <Un} i -1 < max{-7+2-[x - M ( A ) - C M ( B ) ] / M ( A U B ) | 1 < i < n } < £. i Now f o r each 1<i<n, t h e r e e x i s t s d e ( 0 , 1 ) so t h a t i -1 Xd + £(1-d ) i Define f = - 7 + - M ( A ) - C M ( B ) ] / M ( A U B ) . 2»[x i' i as i I(AUB)/2M(AUB) + d .I(C)/4M(C) + (1-d ) - I ( D ) / 4 M ( D ) i where N = function 0\(AUBUCUD) , in A . Note 0 otherwise measure. h + h-l(N)/4S, i would S = /hI(N)dii, that, vanish I t i s easy here on t o check again, E, that a n d h i s some a f arbitrary J*h»I(N)dM > 0 s e t of s t r i c t l y to A belongs 0 for positive . i Now c o n s i d e r 1(A) + c I ( B ) + XI(C) + £I(D) + 7 K N ) . $ We have t h a t J*-f $-f i t 0 /u a.e. du [M(A)+CM(B)]/2M(AUB) - = and i + Xd /4 -1 u = l/2ii(AUB), so Bayesian, i . e . )/4 + 7/4 i ... [2x M ( A U B ) ] = K . i Externally $(\-d i = Write + i that u = x K , i i 1<i<n. Now T is 60 G(*f,/R,,...,^ /K ) n # G ( f , , . .. , f ) n Observe that constant independent and the f the 's are n JG(*f,/K ...,#f /K )dM n n )du l f = /*G(fi,...,f right-hand of the side set n of this (A,B,C,D or evaluated. So, N) (2.4.6) expression on which i n p a r t i c u l a r , the is a both # left-hand i side i s the same whether on G(u/K,,...,u/K ) A or = on B. Hence (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 the equation. Recalling the G(u,...,u) on d e f i n i t i o n of u and both of sides the K of 's, we i find that c*G(x ...,x ) 1 f = G(cx,,...,cx n as asserted i n the statement ), n of the proposition. • n Thus and its the i f a pooling Externally operator Bayesian, corresponding G applied Proposition must be t e c h n i q u e w h i c h we be successfully over the e n t i r e t y of Not a l l homogeneous G's pooling function = max{x G(x) at have u s e d not quasi-local T:A —>A 0 2.4.10 above least to quasi-local tells "homogeneous." reach when © generate operator. is finite an that However, for be for rise could defined example. Externally Consider |l<i<n}, which g i v e s us this conclusion i n c a s e s when G need not n (0,°°) , as i i s both 0 Bayesian instance to the the quasi- 61 local procedure T ( f f ) = max{f f } / / m a x { f f }du. n n n Clearly G i s homogeneous, b u t T Proposition 2.4.11 w i l l Proposition 2.4.11 Let T be a v e c t o r ,...,x y ) . n n now of ones Then i s not E x t e r n a l l y B a y e s i a n , as establish. and w r i t e G(x)«G(y) S>y for = G(x«y)«G(T) the vector f o r a l l x,y (x,y, vectors n in (0,°°) , where G is the function specified in Equation (2.4.3). Proof: Let -1 0 < 7 < min{l,x |i<i<n}, i -1 0 < 6 < min{(1~7)/y ,(x i and let (0,6). 7 A,B,C,D I f we + y M(B) i Next, choose i be d i s j o i n t write e l e m e n t s of J2(M) = r < » so > 0 and i i - 1 - 1 < max{t -[x - 7 - y M(B)]|l<i<n} i - 7 - y n(B)]|1<i<n} 1/x > i -1 «[x i then t w i t h measure i n that -1 X < min{t }, i 1-(y+y u(B)) i i for a l l i = 1 , . . . , n . 0 < X,£ t -y)/y < £. 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 Xd + £(1-d ) i = t i -1 [x i -7-y u(B)]. i i Define f = 7 I(A)/2M(A) + y i + t (1-d i where h e A 1(B) i t d I(C)/M(C) i ( /2)•[hi(N)//hI(N)d ] + )I(D)/M(D) 7 M i i s a r b i t r a r y , and 0 + i is not 0 i s a c o n s e q u e n c e of Now f * 0 ii-a.e. = e\(AUBUCUD). N Lemma (That ThI(N)dM 2.4.8.) and i ft du = 7 + yju i i and hence f e A , + t =1, i 1^i^n. 0 i Consider $ = I(AUB) + /*f du = I(N) 7/2 + + y i XI(C) ju(B) i = 1/x for + £I(D); we have + t d X + t ( 1 - d )$ i i i i + 7/2 1<i£n, i and since <I>f i Externally Equation ^ 0 u~a.e., Bayesian (2.4.6) to we deduce remains may use that constant the the as fact that left-hand the f 's side and $ T is of are i evaluated on A and G( |3x B respectively. |3x ) G(x y,,...,x 1 n i = G(/3, .. . ,/3) where /3 = 7 / 2 M ( A ) . But Consequently, by y we find ) ^ n n G(y ! , . . . ,y ) n Proposition 2.4.10, the left-hand 63 side reduces to G(x ...,x )/G(1,1,...,1), 1 f whence t h e r e s u l t . • n Proof o f Theorem 2.4.6: n Consider is H(x) = G(x)/G(T), a function Lebesgue-measurable that and H(x'y) = H(x)«H(y) on i t follows its conclude to that n w(i) = n x always, i . e . i =1 i H(x) the e x i s t e n c e G(x) defined = (0,<=°) . from P r o p o s i t i o n domain. of n r e a l on By Lemma Then H 2.4.11 2.1.3, we numbers w(1),...,w(n) s u c h n w (i) G(T)• n x i =1 i Therefore T(f 1 f ...,f ) n w ( i) n w ( i) FI f // n f du i=1 i i= 1 i = n The fact that 2.4.10: n L w(i) = i =1 1 x>0 and if Zw(i) G(cx)/G(T) = c follows directly c>0 are n-a.e. from given, Proposition we have n w(i ) • n x i=1 i Iw(i) = c This -G(x)/G(T) completes the proof Summarizing External = cG(x)/G(T). our Bayesianity, "admissible" of Theorem 2.4.6. investigations we have on seen t h a t ^-densities is suitably • quasi-locality provided restricted: and t h e c l a s s of 64 (i) the logarithmic opinion pool n Z w(i) = 1 i s E x t e r n a l l y Bayesian i =1 (ii) opinion i f (0,y) s a t i s f i e s pool i s the only n w( i ) n w( i ) II f d/u /; n f i =1 i i =1 i whatever Assumption with (@,u); 2.4.5, the logarithmic E x t e r n a l l y Bayesian q u a s i - l o c a l pooling procedure a v a i l a b l e . In fact, strengthened, Proposition If {A (0,M) i s n | n>1} opinion this second as we w i l l conclusion presently can be somewhat infinite sequence show: 2.4.12 such of that £l(u) mutually pooling disjoint operator w,,...,w a r e t a k e n n contains (2.4.2) an sets, is then not the logarithmic quasi-local unless t o be n o n - n e g a t i v e . Proof : It suffices a J * f ' ( f / g ) dM t o show t h a t = 0 0 . given a > 0, we c a n f i n d F o r , i f w <0 f o r some f,g e A 0 with i e { 1 , 2 , . . . , n} , l e t a l -w >0 and consider i », Use f = g, f = f , j * i , i j so t h a t a / f . ( f / g ) du - a contradiction. Lemma 2.4.8 t o f i n d a sequence {A |n£l} n of disjoint u~ 65 measurable neighbourhoods, and d e f i n e f for some measure Z K ! I ( A )/[M(A ) i ] + K hl(N)//hI(N)dy i£1 i i = 2 2 h A e zero, 0 in . = 0, then N = 0 \ ( U A ) has i£1 i ( I f fhI(N)dy which case let £ = R, I i£1 I(A )i ] 2 )/[M(A i i instead.) In order 1, so Put that f be i n K i , K > 2 A 0 , i t i s necessary t o have K TT /6 2 1 + K = 2 0 can be chosen a c c o r d i n g l y . g = L,« Z I ( A )/n(k i*1 i i c ) i + L hl(N)/fhI(N)dn 2 where c equals c 2(a+l)/a can > 2. Then g £ 0 M-a.e. and J*gd/i = L, Z 1/i be made equal t o 1 with a p p r o p r i a t e c h o i c e s of L , , L , 2 c 0 < Z 1/i i£1 < J f . ( f / g ) dM ^ a+1 a K, • Z 1 i£1 In t h i s least = last countably necessary. Z i£l [K,/M(A since a+1 c a )i ] - U ( A ) i /L,] *x(A ) i i i 2 • OD. p r o p o s i t i o n , the h y p o t h e s i s many disjoint I f (0,M) i s f i n i t e , the w 's c o u l d not be s t r i c t l y We 2 OD. a Now + L complete this t h a t there be at neighbourhoods i n 6 i s c l e a r l y there i s no reason n e g a t i v e , as long as why some n Z w of =1. s e c t i o n with an example t o show that an 66 E x t e r n a l l y Bayesian operator need not always preserve unanimity. Example 2.4.13 Let e (0,xi) be dichotomous or t a n g i b l e , so that there e x i s t Qiu) with p r o p e r t i e s (I) and ( I I ) . so that min{/j(A) , M ( B ) } > 0. A,,A 2 Write A = A, and B = ©\A, Next, d e f i n e g = 1(A) + I(B)/2 and n l e t T : A —>A be d e f i n e d by T(f ...,f 1 f (note that does not 1/2 < preserve Bayesian. n ) = ftg/JftgdM /f,gdM < 1 s i n c e 1/2 £ g £ 1). Obviously unanimity and i s nevertheless T Externally However, i t i s n e i t h e r l o c a , nor s e m i - l o c a l , nor even quasi-local! 2.5 Information pooling In maximizing and divergence minimizing operators this section, we take a different problem of adequately d e s c r i b i n g a consensus the first approach t o the of opinions. In p a r t , we adopt the p o i n t of view that each o p i n i o n f i contains some " i n f o r m a t i o n " about © and we representative probability distribution, look T ( for a f f single ), whose n expected i n f o r m a t i o n formula so obtained information content w i l l be will differ i s elected. a according maximum. The pooling t o which d e f i n i t i o n of T h i s approach w i l l be seen t o have the 67 merit of p r o v i d i n g a s e n s i b l e i n t e r p r e t a t i o n of the c o n s t a n t s w i with which each o p i n i o n f i s weighted, a question which was i left unanswered p a r t , we employ K u l l b a c k ' s probability formulas pools by our p r e v i o u s attempts. (1968) concept distributions to which c o n t a i n s both as limiting cases. 2.4 to be For convenience, the We we of i n f o r m a t i o n c o n t a i n e d i s the entropy linear begin = the context pooling logarithmic a short review of defined in Section 0 and popular measure of the i n a p r o b a b i l i t y d e n s i t y f on knowledge of 0. e [0,»), i n t r o d u c e d by of communication e n g i n e e r i n g . governed by f) and and the with work on A , measures the " u n c e r t a i n t y " c o n t a i n e d 0 and of which i s b a s i c to the Theory of - J f - l o g ( f )d*x the d i s c r e t e v e r s i o n of which was (as class 0 function E(f) in a between M~a.e.}. {feA|f£0 Perhaps the most c e l e b r a t e d amount of divergence construct Shannon's d e f i n i t i o n of entropy, Information. Then, i n the second The therefore been s u p p l i e d by way in the d i s c r e t e case. Shannon b e t t e r informed Strong quantity E(f) random variable Most our the l e s s u n c e r t a i n one justification of axiomatic (1948) The thus r e p r e s e n t s , i n some sense, s m a l l e r the entropy, the being a p p r i s e d of f . i n the (2.5.1) 0 best is i s deemed to be upon for using (2.5.1) has c h a r a c t e r i z a t i o n s , though only 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 t h a t expected from from the two first (entropy) of the information e x p e r i m e n t s e q u a l s the i n f o r m a t i o n e x p e c t e d experiment plus the c o n d i t i o n a l information the second experiment w i t h r e s p e c t t o the T h i s p o s t u l a t e must be c o n s i d e r e d fundamental to any first. idea of "informat i o n . " The following expression density f in A usually f o r the entropy probability w i t h r e s p e c t t o another p r o b a b i l i t y d e n s i t y g i s 0 known as the K u l l b a c k - L e i b l e r Information d i s c r i m i n a t i n g between f and I(f,g) I t was of one for g: = /f.log(g/f)dn. (2.5.2) d e f i n e d by Shannon (1948) i n the d i s c r e t e case and extended by K u l l b a c k & L e i b l e r quotient log[g(6)/f(6)] evidence" (Good may 1950) or versus H: "the 0 true q u a n t i t y (2.5.2) may be (1951) t o the g e n e r a l c a s e . be i n t e r p r e t e d as the d i s c r i m i n a t i n g i n favour of H,: later the information "the true distribution is regarded the as in "weight © = distribution f." 6 is The of for g" A l t e r n a t e l y , the information gain (a n e g a t i v e q u a n t i t y here) E(f) incurred by using one's - [-/f .log(g)dix] "best knowledge of ©," g, t o take 69 decisions, while distribution Two are the true (hypothetical) underlying probability governing 0 i s f . basic properties of the K u l l b a c k - L e i b l e r information embodied i n Lemma 2.5.1 Let I:(A ) —>R represent (i) be d e f i n e d 2 0 any n o n - v a n i s h i n g by E q u a t i o n (2.5.2). ^-densities in A , 0 If f and g then I ( f , g ) £ 0 a l w a y s , and (ii) I ( f , g ) = 0 i f f f=g *i-a.e. Proof: This Chapter 2 of Kullback The for a is stated (1968). of are enough suggest opinion pool moment probability that to and the assessments Information a new c h a r a c t e r i z a t i o n o f t h e decision f a s Theorem 3.1 i n Kullback-Leibler i n the f o l l o w i n g a proved • above p r o p e r t i e s measure linear result context. maker f Let us imagine has c o l l e c t e d n about 0 and that expert he is n informed, density as a knows ( i ) one o f t h e s e i s the 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 random variable; and ( i i ) t h e p r o b a b i l i t y t h a t 0 the i - t h n f , i s objective i s p £ 0, I p = 1. We have i i i=1 i r e m a r k e d i n C h a p t e r 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 distribution, already o r j u d g e s somehow t h a t 70 0 may only instance), be so hypothetical. If f according adopting virtual the (© may situation be o b s e r v a b l e which we However, i t i s s u g g e s t i v e are o n l y once, f o r describing is and d e s c r i p t i v e . 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 t o (2.5.2), the amount of information lost 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 due to would be i -I(f i ,g) = J f - l o g ( f /g)dy. i i A v e r a g i n g over the f ' s , we i mation l o s s f i n d t h a t the g l o b a l e x p e c t e d is n - L p - I ( f ,g), i=1 i i a functional depending choose g so as t o distribution occasioned of randomly s o l e l y on g. minimize which (2.5.3) (2.5.3), operator chosen I t would seem n a t u r a l t o i . e . pick a probability m i n i m i z e s the e x p e c t e d l o s s of by the need t o compromise. pooling infor- from Note t h a t the information definition r u l e s out the p o s s i b i l i t y t h a t g c o u l d the f 's: although attractive, be this i selection scheme does not engender the idea of consensus. If a n pooling operator T:A 0 —>A 0 is such that T ( f f ) n = g 71 minimizes (2.5.3) whatever be f,,...,f ((p 1 r ...,p n fixed vector Leibler probabilities), Information Theorem The of Maximizer we say ) being a n that it is a Kullback- (KLIM). 2.5.2 linear opinion pool T(f moreover, w = p i n Z w f i=1 i i f ) = n i s the only KLIM; , i=1,...,n. i Proof; Call f n Z p f . i =1 i i = equivalently, Lemma 2.5.1 outset. the = f ji-a.e. • have a c h a r a c t e r i z a t i o n impose a s p e c i f i c Locality definition a KLIM, g must m i n i m i z e n Z p /f -log(g/f)dM i =l i i shows t h a t g = which does not d o e s not be (2.5.3) o r , maximize H e r e , we the To of T. 2.5.2 weights, w of the linear form on the pooling operator comes A l s o noteworthy distinguish Theorem merely between provides , at least I(f,g)• as a i s the t a n g i b l e and opinion consequence fact that intangible this of at the result spaces. us with a natural interpretation as they appear i n the pool linear of opinion i pool. If an objective probability density, f, for 8 and 72 objective w probabilities, should p , o f {f =f} e x i s t , i i e q u a l p , 1<i<n. i i unknown, When f exists i t would seem n a t u r a l t o l e t w we have s e e n but the represent that p 's i are 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 the "right the one." that the i - t h expert T h i s supports the i n t u i t i v e a b s e n c e o f an o b j e c t i v e d i s t r i b u t i o n , should the be c h o s e n determining (1968) the accuracy the weights. at (1965), looks ability likelihood of the experts; B a y e s ' Theorem t o f o r m a l l y the and on the this revise related weights rules for {w suggested to by compare t h e i n v o l v e s the a p p l i c a t i o n the observation. the weights a f t e r produce a s e t of w , i assessor. one, ratios d e c i s i o n maker c o u l d u s e t h e p r e s e n t consensus weights, A l l o f them a r e b a s e d on t h e i n t u i t i v e Roberts assessment the d e s c r i b e s some o f t h e most p o p u l a r The most p r o m i s i n g of i d e a t h a t even i n of each grounds proposed above. predictive is on t h e b a s i s o f a s u b j e c t i v e judgement made by d e c i s i o n maker c o n c e r n i n g Winkler f, opinion having weights More s i m p l y , methods asked |i£j<n} after on to each though, extract each e x p e r t , the basis a i ,to of the i j relative importance t h a t he w o u l d a s s i g n t o t h e o p i n i o n s o f t h e v a r i o u s members o f t h e p a n e l , this raises pooling questions about t h e w e i g h t s and t h e v a l u e particular go further including weights assessment. on f o r e v e r , except himself. the formula (weight) course, t o be u s e d i n t o be a s s i g n e d In p r i n c i p l e , that the f i n a l Of t h i s process consensus will t o any could generally 73 be less the s e n s i t i v e to pooling (1964, p. modest the choice formula. 264), 'this changes "In a i s an in of weights than way,' old choice state Mosteller story weights to the in & Wallace statistics ordinarily change of because the output modestly." The idea information as a but rather the context expert object of to gain various Bernardo may the sensible any a parallel that of have kind seen Fisher (1934), by the idea has of when and the the a quantity linear statistic, "summarises t h e sample." decisions is precisely here: collecting it is the to take decisions. maximization procedure is (Bayesian) p r i n c i p l e that i t i s , i n some object i s t o make In Chapter a f r o m a number of pool may a statistic whole o f designs likelihood determining opinion (1956) relative not a p p l i c a t i o n i n mind. p r o b l e m of It e x p e r i m e n t , and general utility, Lindley reach world." nature, this the to been a p p l i e d an s t a t e s of one the i s not the by for experimental problem to assess shown t h a t between t h e sufficient supplied our specific that criterion v i e w e d as expected estimating of be instance the without hoc) Kullback-Leibler been s u g g e s t e d experimentation possible maximizing only ad expected knowledge a b o u t (1979) has another the I t has i n which t h i s p u r p o s e of the but new. (but opinions stated maximizing i s not sensible "where t h e of of the be of sense, inference 1, we drew consensus and observations. We i n t e r p r e t e d as which, according relevant a to information 74 It would so-called seem n a t u r a l Renyi I n f o r m a t i o n measures -1 I ( f , g ) = (1-a) a based t o t r y t o e x t e n d Theorem 2.5.2 t o t h e on t h e g - e n t r o p y a 1 -a .log[Jf g dii], I a by Renyi ( f , g ) —> I ( f , g ) restriction a < (2.5.4) i s a l w a y s we would like -1 a - l o g [ J f dn], (1961). i s imposed finite. to find 0<a<1 A s t h e r e a d e r may e a s i l y a s a—>1 w h a t e v e r 1 (2.5.4) functions E ( f ) » (1-a) a introduced 0<a<1 be f a n d g i n A; t o ensure Reasoning a possibly that here, the the i n t e g r a l i n i n t h e same way a s unique check, g = P ( f f a before, ) which n maximizes n Z p I (f ,g), i=1 i a i the expected problem Renyi Information h a s n o t y e t been a solution solved of (2.5.5) o r d e r a, 0 < a < 1. for arbitrary f o r t h e c a s e where n=2 a n d a = (2.5.5) becomes l o g { [/v/I7gdti] [ Jt/IIgdii]} n a n d a. = p i s given 2 This However, =1/2 below. where 75 Lemma 2.5.3 The [ jVITgdM] quantity H /fH du P r o o f : L e t F, = /IT, F achieves jVf"7gclri] M-a.e., H = /IT 2 2 [ + a maximum /II. = y/Tl, where H = F, + F 2 when g = 2 and G = / g . We have 2 JF iGd/u SF Gdu < 2 and The second J*F Gdu 0 inequality with particular Sharpe in that 2 * [/(F,+F ) du]-[/G dM] = /H dy. 2 2 2 2 zero unless (Rudin 0(H) partial setting 1974, p . 6 6 ) . ... = w inequality 2 * for then 0 0 and g = M~a.e. i s also achieved It by this • result was o b t a i n e d communication). o f Example = 1/n. u~a.e. If 7=0, Thus 7 2 the f i r s t in general, P = yg 2 T h u s /3 * 0 and g = H / / H d * i c h o i c e o f g. (private the that, = 2 2 is strict Jgdj* = 1. happens This [ J(F,+F )GdM] so H = 0 j/-a.e., a c o n t r a d i c t i o n . and 0H2/7 ^ 2 some 0 , 7 e R, n o t b o t h so + [/FjGdiu] 2 so 4 J"F ! GdM # [;F,Gdy] independently I t i s he who p o i n t e d o u t 2.5.5 b e l o w , 2 The d e t a i l s that, i t i s p o s s i b l e t o show n n * ( I v T"') //( I /T~) dM a i=1 i i=1 i / by Mr. B . J . 2 f o r n > 2 and w, are omitted, n We now propose o p e r a t o r s w h i c h we c a l l to characterize the "normalized the following (weighted) pooling means of 76 order a:" T (f,,...,f ) = a n (As before, the n a 1 /a n a 1 /a [ I w f ] //[ I w f ] du, 0<a<1. i=1 i i weights (2.5.6) i=1 i i w are non-negative and sum up t o 1.) i These bear a, M ( a an o b v i o u s c o n n e c t i o n w i t h t h e w e i g h t e d x x ), o f a s e t o f n n o n - n e g a t i v e & P61ya 1934): M(x,,...,x) a n weighted discuss means t h e problem The notion numbers ( c f . n Hardy, L i t t l e w o o d These real mean o f o r d e r basic will appear of p o o l i n g quantity, n a 1 [ Z w x ] i=1 i i = again /a i n Chapter 3, when we propensity functions. here, is Kullback's o f d i v e r g e n c e between any two p r o b a b i l i t y (1968, p . 67) distributions, f and g: 6 (f ,g) a In t h e c a s e Hellinger = a - 1/2, 6 ( f , g ) a (1909) - K a k u t a n i a 1-a 1-Jf g is du, 0<a<1. equivalent (1948) - M a t u s i t a to the so-called (1951) d i s t a n c e 77 P (f,g) - 2 a measure (1965) f o r measuring the d i s t a n c e distributions distributions. to as the a f f i n i t y We 2 a l s o used by S t e i n between p o s t e r i o r prior J ( / r - /g) d>, The obtained from two different f u n c t i o n p(f,g) i s sometimes r e f e r r e d between f and g, a f t e r Bhattacharyya (1943). have the f o l l o w i n g Theorem 2.5.4 The p o o l i n g operator T d e f i n e d by Equation (2.5.6) i s the the expected divergence n I w 5 (f ,g). i=1 i a i only a one which minimizes Proof: n a I w f . i=1 i i Write f = By Holder's inequality, 1-a Jfg 1/a dn * [Jf a d/u] < » 1/a and equality i s achieved only when /3f /3,7 e R, not both zero (Rudin proof of Lemma 2.5.3. If a decision opinions f f 1974, = yg p. 66). u-a.e. Proceed as f o r some in the • maker knows t h a t each one of the n expert which he has c o l l e c t e d has a corresponding n probability p i of being the " r i g h t one," then i t might w e l l seem 78 reasonable to him t o choose a consensus d i s t r i b u t i o n which, on the average, w i l l distribution. have the g r e a t e s t In that "affinity" with the true case, Theorem 2.5.4 above says that T a should be used f o r some 0 < a < 1. may be guided by decision maker conclusions the might specific want to The c h o i c e of the value application; assess a l t e r n a t e l y , the the s e n s i t i v i t y the linear and the i n c l u d e d as l i m i t i n g cases. a —> of h i s by computing a consensus f o r d i f f e r e n t a's. One a t t r a c t i v e f e a t u r e of the c l a s s {T } i s the both of a 1, whatever f f logarithmic Indeed, T ( a e A. pooling f f On the other fact that o p e r a t o r s are n ) -> Z w f n i=1 i i hand, we can use n L'Hospital's log[Lim a—> 0 r u l e t o see that n a 1/a -1 n a ( Z w f ) ] = Lim a «log[ I w f ] i=1 i i a—> 0 i=1 i i n a n a = Lim [ Z w f . l o g ( f ) ] / [ Z w f ] a—>0 i = 1 i i i i = 1 i i = log[ Now, n w(i) f ]. i =1 i n & P6lya i t i s known (Hardy, L i t t l e w o o d n 1/k k [ Z w f ] i=1 i i £ n Z w f i=1 i i as 1934, p. 26) that 79 pointwise for a l l k £ 1, and of n / Z w f du course = 1. i =1 i i Therefore, (Sion we c a n u s e t h e L e b e s g u e D o m i n a t e d C o n v e r g e n c e 1968, p. 95) t o c o n c l u d e Theorem that 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 ) f du. i =1 i / n Consequently, This fact n w(i) n w(i) T —> n f / / n f du a i=1 i i=1 i may Externally provide Bayesian procedure We c o n c l u d e t h i s Weerahandi Example & Zidek that members, e a c h votes. Mr. indication a section with an a —> 0. quasi-local sense. borrowed from composed of n (1978): House of procedures) Representatives also that when f o r a p p r o v a l , e a c h member say, this example is o f whom h a s a d e m o c r a t i c w e i g h t Suppose Speaker that as i s " r o b u s t " i n some 2.5.5 ( P a r l i a m e n t a r y v o t i n g Suppose House some always his 1/n when he a p r o p o s a l i s p u t b e f o r e the i tells personal of an independent probability 0 < p judge, < 1 that i passing the understanding consensus proposal is that is the this and t a k e a d e c i s i o n , right person thing is to required a p p r o v a l or r e j e c t i o n , do. The t o form t h e which is 80 consistent to vote with i t . Note t h a t Mr. S p e a k e r c o u l d have t h e r i g h t t o o , a s l o n g a s he does n o t influence capacity unduly the as a r b i t r a t o r decision l e t h i s personal ultimately of the group. desires made by him i n h i s (0 I f Mr. S p e a k e r u s e s T a < a < favour will 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 of passing the proposal ( o n c e he has h e a r d every deputy) be n a 1/a n a 1/a [ I p ) / [ Z (1-p ) ] , i=1 i i=1 i n [ n p ] i=1 Thus, the passing best 1/n n /[ n ( 1 - p ) ] i i=1 non-randomized 1/n , a = 0. decision rule would c o n s i s t of the proposal i f (1/n)- n Z l o g [ p /(1-p )] > 0, i=1 i i When a = 1, t h e p r o c e d u r e When a = 0, t h e average, 0 < a £ 1 i n a n a (1/n)• Z p > (1/n)• Z (1-p ) , i=1 i i=1 i 1/2. odds i n the 0 < a S 1; a = 0. reduces t o passing proposal parliamentarians' will (2.5.7) go (2.5.8) the proposal through i fp > i f , on log-odds-ratios favor the passage, 81 i.e. (2.5.8) Now holds. suppose t h a t against passage, assume t h a t of 0 and = 7 < p n the the 1/2, House other members, n, i=1,...,n , = n-n and 0 ...,n. Weerahandi & Zidek (1978) p o i n t non-randomized d e c i s i o n r u l e familiar "simple (2.5.7) Thus T majority" voting for a l l 0 < a £ i s not 0 = I > are 0 for. Further 1/2, i=n +1, 0 i optimal of are 0 p i i=1,...,n extraordinary majority" rule politicians were as a p p e a r a n c e s and would 1, therefore the This reader may respect. well they p if 7 = (2.5.8) i s n o t h i n g the in this c e r t a i n as that procedure. as seem out the but the is also easily But at to be true check. the justified, appear 1-£, "simple least in 's were a l l e s s e n t i a l l y if public 0 or 1. i 2.6 Discussion The the where respect T,; as the previous t h e o r e t i c a l aspects case (i) work of to a proposed and the regard problem. expert fixed of opinions "only the are arguments ( i i ) characterized Some of the result been d i r e c t e d expressible as the densities with In p a r t i c u l a r , favouring the the logarithmic as following our linear pooling Bayesian main remarks toward in measure. practical" ' Externally latter has group p r o b a b i l i t y assessment underlying new sections we have opinion pool, operator, T , 0 procedure. We c o n t r i b u t i o n to this will also apply in 82 substance to the developments of Chapter clear, the 3 (they will not be repeated). It should especially, problem of reached linear is that the by or be there panel (1975). logarithmic i n t e r e s t i n g to think entire c l a s s of personally the appealing The of pool of the reasonable favour axiom cannot the in a Bayesian of unique is i s by two now as the being use of e i t h e r limiting formulas. This operator, Madansky 2.5 the conclusion well-justified, pooling of Section s o l u t i o n to also C e r t a i n l y , the pooling EB a This logarithmic coherence content be of e x p e r t s . Bacharach the from and c a s e s of the it an author would as finds he (1964;1978) rather framework. prescription n w ( i) n w (i ) n f // n f du i=1 i i=1 i is also recommended by by Morris (1974;1977) and natural-conjugate that offered probability conjugate valuable well Weerahandi by (N-C) the assessments family only of t o the Winkler & Zidek The logarithmic that N-C opinion belong distributions. extent ( 1 9 7 8 ) , and (1968), the approach. must (2.6.1) to Of latter recipe pool some course, Bacharach through his amounts to except that a l l fixed natural- this such a mathematical a p p r o x i m a t e one's judgements. indirectly approach model is may (1973) a t t r i b u t e s 83 (2.6.1) t o Hammond, b u t he does not cite a source f o r the result. As Winkler operator (1968) points out, by practical c a n be i n f l u e n c e d instance, the desire to from using procedure, relevance (i) opinion sensible features pools i n the context Ti i s generally For formula. well deter Morris' (1977) assessment problems applications. the f o l l o w i n g linear pooling considerations. f o r example, e n t a i l s f o r m i d a b l e Thus, of a f o r t h e c o n s e n s u s may an o t h e r w i s e in a l l but the simplest and choice s i m p l i f y c o m p u t a t i o n s o r t h e need t o have an a n a l y t i c a l e x p r e s s i o n one the of T a n d T,, t h e l o g a r i t h m i c 0 respectively, would be of some o f an a c t u a l a p p l i c a t i o n : multi-modal, whilst T 0 i s typically uni- modal It amongst the is generally observed t h e modes o f t h e i n d i v i d u a l more likely i t is distribution. The f a c t which t y p i f i e s i t s modes well for that be p e r c e i v e d a decision that that T t T, may ( i . e . the as a f a u l t , ( c f . Weerahandi even the l a r g e r probability will fail the d i f f e r e n c e s densities f , i produce a multi-modal to identify individual a parameter choices) might i f the problem does not c a l l & Zidek 1981). 84 (ii) has a g r e a t e r v a r i a n c e than T 0 T h i s i s not s u r p r i s i n g i n view of w 's, i Given under T multiplicative than 0 analysis under nature analogy, t h i n k of the where T,. set Whether This is due to of the l o g a r i t h m i c o p i n i o n p o o l . situation faced in a formal a l a r g e amount of sample i n f o r m a t i o n r e l a t i v e l y smaller amount of p r i o r of a small or a l a r g e v a r i a n c e N(0,1) and f "swamps" a i s more d e s i r a b l e * N(1+/3/e,e ) 2 2 For an Bayesian on the p a r t i c u l a r a p p l i c a t i o n one has i n mind. f, the knowledge. (1976) r e p o r t s the f o l l o w i n g example: suppose that gave a the t i g h t e r d i s t r i b u t i o n s w i l l a u t o m a t i c a l l y r e c e i v e more weight depend (i). Bernardo two as t h e i r will experts respective o p i n i o n s , and that 8 l a t e r turned out t o be 1. Figure 1. Two o p i n i o n s with a d i f f e r e n t entropy but g i v i n g the same p r o b a b i l i t y t o the true value of the q u a n t i t y of i n t e r e s t before i t i s r e v e a l e d to be 1 . 85 Explains loose the Bernardo, approximation grounds that approximate possibly fatal needed." our present (iii) We the be On the have decision and a other fatal grounds "In useful, attaches e r r o r may on is 6 may it values. where a s m a l l preferred of page 3 4 : on that a f sample could 2 s u r v e y where a be preferred high probability hand, in a medical consequences, f, i t warns a g a i n s t rather suggests that These p r e o c c u p a t i o n s , however, a r e to such research could a on be premature, more evidence somewhat beyond concern. C a l c u l a t i o n s are remark t h a t if f easier f with are T 0 members of a family of n exponential type then T ( f if f i then T 0 ( f by the same g e n e r a l i z e d density, f ) w i l l be a member o f t h e same family. n i s a n o r m a l d e n s i t y w i t h mean u and v a r i a n c e i 0 example, determined f ) will also be and variance 2 form = n 1/ Z a , where a i=1 i i n Z w f i =1 i i Distributions of usually i n t r a c t a b l e , unless are the o of are = w /a , i i called c o u r s e a l l the 2 's a r e i same. 2 1<i£n. mixtures f a , i n Z a u / i=1 i i n o r m a l , w i t h mean u = n n Z a i=1 i For and the 86 Professor logarithmic that A.W. opinion i t w o u l d be when these sources. where This i s not was obtained as a criticism are comes the EB logarithmic ought not overlapping a by-product the pool. be a well applied to data approach, opinion I t may or rule. However, pool interpreted logical out opinions experience be the pointing expert o f Bayes* I t must postulate, to W i n k l e r ' s N-C logarithmic 2.4.6. to objection combining a problem with of an communication), for b a s e d on i n Theorem of raised (personal as a criticism i s the has unsatisfactory i s indeed (2.6.1) postulate pool opinions this which Marshall which instead consequence be that the T and set {f of the EB } for the collection of i reasons c i t e d opinions derived specified, EB Morris the function from t h e will evaluation empirically determining of around t h i s experts, calibrating C will be the rather experts In the well this encoding as each calibration maker's the rendered d i f f i c u l t by is in a so-called Joint decision than as general, themselves; e l u s i v e d e p e n d e n c i e s between difficulty ability —>[0,°°). represent the the assessment C:© alternate 's. However once t h e d e r i v e d s e t i lead to t h e i r logarithmic pool. d e p e n d e n c e amongst Function t o an f (1974,1977) g e t s probability Calibration instead would a g a i n d e g r e e of expert's the a b o v e , but result and by so the experts' subjective of the the need experts task to opinions. of assess 87 When t h i s i s done, a " g e n e r a l i z e d l o g a r i t h m i c o p i n i o n p o o l " M(f n ) emerges f o r t h e composite p r i o r of n e x p e r t s : M(f,,... f ) n f This n n C n f /JC n f du. i=1 i i=1 i = (2.6.2) reduces t o (2.6.1) w i t h w ( i ) = 1,1<i<n, i f t h e e x p e r t s a r e independent concurs and c a l i b r a t e d ( i . e . i f C is a constant). n w i t h t h e ad hoc s u g g e s t i o n of W i n k l e r (1968) t h a t E w i=1 i s h o u l d be taken i n t h e i n t e r v a l [ l , n ] and r e f l e c t of independence" Externally variance between Bayesian than T and the e x p e r t s . will characterize "amount M i s not usually have a much smaller i f C i s a c o n s t a n t ( c f . remark ( i i ) 0 pooling the However, A l s o , note, t h a t M p r o v i d e s us w i t h an example of a local" This operator. It a l l the E x t e r n a l l y would be Bayesian above). "semi-quasi- interesting to semi-quasi-local pooling operators. As observed by Winkler (1968) and (1981), p o o l i n g o p e r a t o r s can be used i n an spirit from assumption Weerahandi entirely t h a t which has m o t i v a t e d t h e i r s t u d y . t h a t each f represents a subjective & Zidek different Indeed, t h e probability i distribution assessed by a member of a p a n e l of e x p e r t s i s c o n v e n i e n t , but not c r u c i a l t o t h e a n a l y s i s . individual may well Thus, a single choose t o r e f l e c t t h e surmised q u a l i t y of 88 his p r i o r knowledge i n an a n a l y s i s by combining "mechanical predictions" representing tendency to p e r s i s t e n c e (what today), or else derived multiple regression. a p p r o p r i a t e weights Finally, a s p e c t s of assumed a our that candid and staff the problem Throughout in their of as determining this game-theoretic chapter, we have probability assessments. (1968), we c o u l d say that they are d e d i c a t e d whose s i n c e r i t y i s unquestionable and who c o n s p i r e to t r i c k the d e c i s i o n maker. assumption happen e x p e r t s c o n s u l t e d by the d e c i s i o n maker were accurate men, will complex schemes such of c a u t i o n concerning the problem. Paraphrasing R a i f f a or a remains. word the ignorance yesterday more Nonetheless, with either happened using this prior T h i s i s not a would not realistic i f the u l t i m a t e o b j e c t i v e of the e x e r c i s e i s to make a ( p o s s i b l y c o n s e q u e n t i a l ) d e c i s i o n and/or i f the d e c i s i o n maker only r e p 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 the example, opinion one in particular Fellner of an experts attempt to may intentionally influence the falsify his others in discusses group probabilistic decision "slanting" making. When i n v o l v e d , the s o l u t i o n s presented here w i l l prove u n l e s s , perhaps, same For toward a consensus which i s somehow advantageous to h i m s e l f . (1965) occurrence the group. the members of the panel preference Weerahandi & Zidek pattern, its bargaining to function. the is unsatisfactory share . (roughly) i.e. utility (1981), our approach and the Unlike multi-Bayesian d e c i s i o n problem i s through a g g r e g a t i o n , 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 reconciling have judgemental concentrated probability on the problem assessments, of i . e . expert o p i n i o n s which a r e e x p r e s s i b l e as d e n s i t i e s w i t h r e s p e c t t o some n a t u r a l dominating alternatives. concern In measure on a space 9 this chapter, mutually exclusive e n l a r g e t h i s problem and o u r s e l v e s w i t h what we c a l l p r o p e n s i t y f u n c t i o n s , or functions for short. Given a space, 0, of contemplated of n a t u r e , a P - f u n c t i o n i s j u s t (0,»). we of a transformation of states 0 into We w i l l denote by n t h e s e t of a l l P - f u n c t i o n s on 0. Examples of P-functions are l i k e l i h o o d s , belief functions (Shafer 1976) and d e n s i t y f u n c t i o n s such as those o b t a i n e d prior, vague prior, fiducial distributions. posterior, structural These f u n c t i o n s , p, f i n i t e i n t e g r a l w i t h r e s p e c t t o any p a r t i c u l a r they share from ( F r a s e r 1966) and need not have a measure; however, the property that p(0)/p(77) represents the r e l a t i v e degree of support over P- (or "propensity") expressed 17, 8 and v b e i n g elements o f 0. i n favour This r a t i o w e l l be an o d d s - r a t i o o r a l i k e l i h o o d - r a t i o , of p(t9)/p(7?) f o r example. 8 may In any c a s e , t h e l a r g e r t h i s q u a n t i t y i s , t h e g r e a t e r i s t h e degree of c o n v i c t i o n i n f a v o u r of 8 compared t o T J . 90 To f i x i d e a s , some s p e c i f i c where the need applications will will occur also examples w i l l to serve pool now be presented P-functions. as a motivation These f o r the ensuing n developments. In each case, a pooling operator T:II — > n is required. Example 3.1.1 (pooling u t i l i t y When c o n f r o n t e d w i t h necessary to Typically, this will be made panel (or take which w i l l functions) intra-group c o n f l i c t s , into consideration be t h e c a s e affect consequences) q u e s t i o n s of t h e members of t h e p a n e l . assignments a s w e l l a s on t h e p r i o r for the p o s s i b l e s t a t e s of nature, a decision to decompose two the probabilities, matters and problem into proceed independently. to This attitude sought through decide on functions. no avail one. a that formula In t h a t a the for I f such a for actions probabilities choose utilities consensus i s recommended and on by both Raiffa shortcomings. to this decision amalgamating respect, the techniques because u t i l i t y is solution aggregation, On t h e o t h e r function a a i s to maker m i g h t parts, extract (1968, p . 232) d e s p i t e some o f i t s Assuming utility. i f a choice or d e c i s i o n o f e x p e r t s d i s a g r e e s on u t i l i t y their i t i s sometimes consensus problem i s maker the will have experts' of Chapter to utility 2 are of f u n c t i o n s g e n e r a l l y do n o t i n t e g r a t e t o hand, n o t e P-function, t h a t any s t r i c t l y so that positive - i n that case utility at least- 91 pooling utility Furthermore, the attitude to cardinal), monotonic Example 3.1.2 that conditional likelihood i s 6} the statistical convenient way to (but functions can applications, unknown) more underlying modelling involves arbitrariness the of facts part strictly not a l l such principle). derived from the form of 8. of These F o r , i n a l l but they a r e modelled or less i n some accurately the d i s t r i b u t i o n of the data. some the a r e a r e f l e c t i o n of that a p r o b l e m were d e l i c a t e not be s u r p r i s i n g on prior not i n agreement where a (as introspection assessor, mild and and subjective "interpretation" of and e v i d e n c e . If piece are be s u b j e c t i v e . This mathematical likelihoods to course, distributions approximate on adopt i s ordinal considered as a function distributions simplest (of up i f we likelihoods) conditional observed a measure o f u t i l i t y probability P{data|true value P-functions. s a t i s f y the e x p e c t e d - u t i l i t y (pooling general, pooling a s s u m p t i o n c a n be w a i v e d transformation functions w i l l to and h e n c e u n i q u e o n l y increasing utility In amounts the p o s i t i v i t y Paretian opposed functions that a panel probabilities either of information the data and c o m p l i c a t e d enough, o f e x p e r t s who were n o t i n a c c o r d f o r the possible states of nature on t h e meaning a n d / o r v a l u e presented was r e l i a b l e , i t would t o them. At best, o f good q u a l i t y o f some i n those and r e l a t e d were new cases to the 92 parameter of i n t e r e s t critical experts significance their and (common) conceive of probably might even adopt "revised to where in i n dissensus, However, on their i t is easy to t h e new i n f o r m a t i o n would be s o interpretation expert even a f t e r agree even the a s s o c i a t e d l i k e l i h o o d as to disagree which i s well understood, nearly opinion." personal the experts circumstances left would situations much s u b j e c t would cause i n a manner w h i c h even that i t s disclosure further! probability This suggests a s s e s s o r s c o u l d be an open a n d v i g o r o u s exchange of informat ion. One about method the i n t e r p r e t a t i o n associated t h e d i s a g r e e m e n t amongst of a s e t subjective likelihoods most for resolving of likelihood data is to functions. pool their When these a r e n o n - z e r o e v e r y w h e r e on 0 ( t h e u s u a l c a s e interesting one), this reduces to the experts and t h e problem of aggregating propensities. Example 3.1.3 This (combining independent for a general bibliography on formal tests of hypothesis) i s a w e l l - k n o w n p r o b l e m w h i c h h a s been many a u t h o r s ; The 1 1 the statistical We are thankful to (London) f o r s u g g e s t i n g d i s c u s s i o n and s u b j e c t , we r e f e r a i n v e s t i g a t e d by fairly t o Monti extensive & Sen (1976). p r o b l e m may be s t a t e d a s f o l l o w s : g i v e n n Dr. Peter McCullagh of I m p e r i a l this application. College 93 independent test statistics, J 1 f ...,J , for testing a null (0 = ft U&i b e i n g a space of n hypothesis H :weS2 probability distributions), 0 versus 0 H^rcjefl, select 0 a function, P, of J 1 f ...,J n which is behind the construction tests, to to be u s e d a s t h e combined possibly of m a r g i n a l scientifically viewed as of P i s that decisive a whole. test the significance conclusions statistic. aggregate The i d e a of individually, if their several can lead results are I f l a r g e v a l u e s o f t h e J 's a r e c o n s i d e r e d i critical for testing H , 0 one common s o l u t i o n a L consists of finding t e s t P b a s e d on t h e o b s e r v e d s i g n i f i c a n c e l e v e l s o r P-values, 1-F ( J ), where F ( t ) = P ( J <t}, t h e c u m u l a t i v e i i i i i d i s t r i b u t i o n of J under t h e n u l l h y p o t h e s i s ( i t i s assumed t h a t 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 t h e same f o r a l l w e i 0 fi )• For 0 P(L 1 f example, F i s h e r ' s ...,L ) n 1/n n L i =1 i = n value and H i s rejected 0 i s g i v e n by when t h e o b s e r v e d of P i s small. DeGroot the (1932) omnibus p r o c e d u r e tail (1973) h a s shown t h a t area L as a p o s t e r i o r i t i s possible probability to interpret or as a likelihood i ratio f o r the acceptance of the n u l l . Because o f t h i s , each L i may be function regarded as an individual assessing the " l i k e l i h o o d " expression o f 0 = {"H 0 of b e l i e f is true"} o r P(the 94 uninteresting c a s e where L = 0 being neglected). In that case, i n P(L 1 f ...,L ) i s a pooling operator acting n = (0,1) . on A n Example 3.1.4 (The B e r g s o n - S a m u e l s o n Economists individual's (cf. basis the a distinction social-welfare Samuelson quantifies, the make 1947, function Chapter of t h e i r personal hand, the basis the utility function A person's u t i l i t y i n t e r e s t s o r on any o t h e r social-welfare of impersonal differently, the individual's "ethical" It social is function or "moral" describes their correspond vector (u,,...,u ) of a "social private would On prefer) alone. preferences, Stated whereas "subjective" a social-welfare basis. represents the former which t h e person Mathematically, makes rather, function on i s supposed t o considerations were c a l l e d upon t o make a m o r a l v a l u e which function (or, social-welfare function pattern. his an e i t h e r c a r d i n a l l y o r o r d i n a l l y , what t h e y p r e f e r other utility function) between what t h e y c a l l and 8). e x p r e s s what t h e i n d i v i d u a l p r e f e r s on social-welfare the the preference would u s e i f t h e y judgement. function utility" utilities. is a mapping W W(u,,...,u ) t o any n Provided that its n domain of and range a r e a p p r o p r i a t e l y a P-function determining pooling the operator. r e s t r i c t e d , W i s an The q u e s t i o n form o f a " r e a s o n a b l e " instance of d e f i n i n g social-welfare and function 95 has occupied Generally should economists speaking, be should associated with increasing "if i . e . the s o c i a l only depend role on the we s h a l l 1955). points: (i) W level attached individual prospect; i n each of i t s arguments, t h e then Harsanyi two utility on that p a r t i c u l a r utility, As (cf.,e.g., i s agreement y o u i n c r e a s e any a g e n t ' s u t i l i t y else's to a utilities a n d ( i i ) W s h o u l d be rationale without being decreasing that anybody s o c i e t y i s made b e t t e r o f f . " see, these two conditions play an important i n the sequel. Def i n i t i o n A there "local," "prospect" f o r some time pooling 3.1.5 n T:I1 —>IT operator is called local whenever there n exists a f u n c t i o n G:(0,°°) — > ( 0 , « ) T(p,,...,p a l l 6e& and p 1 f that )(6) = G ( p , ( 6 ) , . . . , p n for such (0)) (3.1.1) n ...,p e II. n Note t h a t E q u a t i o n (3.1.1) must h o l d merely "almost absence o f any n a t u r a l c h o i c e The following operator t o be everywhere." This i s rendered f o r a dominating lemma g i v e s o b v i o u s local. everywhere, and necessary measure not by t h e on ©. e q u i v a l e n t c o n d i t i o n s f o r an 96 Lemma 3.1,6 A pooling (i) T ( p 1 f n T:II — > operator — II i s l o c a l i f f ,p ) ( 0 ) = T ( p , , . . . , p n all (0) = p )(TJ) whenever p n i (TJ) f o r i i=1,...,n; and (ii) T(p,,...,p )(0) = T(q,,...,q ) ( 0 ) whenever n n a l l i=1,...,n. for Proof: This is trivial. Condition the experts "equally maker both in likely," should 0 and TJ. this To assume sense believed that the hope o f g a i n i n g can as be r e g a r d e d before Definition and according attribute is be c a l l e d states 0 then, particular (0) = q (0) i i • ( i ) above c o u l d agree t h a t p TJ "consistency." are to Condition t h e same " l i k e l i h o o d " that If a l l "equiprobable" ( i ) , the d e c i s i o n or " p r o p e n s i t y " t h e d e c i s i o n maker is d i d not b i a s t h e i r some s t r a t e g i c as a l i k e l i h o o d advantage. principle to consistent seems n o n - c o n t r o v e r s i a l , a t l e a s t the assessors or if i t judgements i n Condition for P-functions, (ii) just ( c f . page 2 9 ) . 3.1.7 n A pooling operator T:I1 — > b e l i e f s (POB) i f f ( i ) T ( p . . . , p ) ( 0 ) <. T ( p n 1 f II i s s a i d 1 f to preserve the o r d e r i n g of ) ( T ? ) whenever p ...,p n (0) < p i (TJ) i for 97 all i=1 ,...,n; and (ii) T(p,,...,p ) ( T J ) whenever )(0) < T(p,,...,p n for a l l i=1,...,n The set above o u t by property economists f o r the considerable often acts as a " r e g u l a r i t y POB pooling In with large four 3.2 requirements of satisfy will i n Chapter will of r a t i o n a l i t y P-function appear be 2. Apart find that i t similar to Note t h a t any ( i ) o f Lemma used in 3.1.6. conjunction in order to c h a r a c t e r i z e pooling the will manner Condition below, l o c a l i t y postulates class used in a requirement function. a p p e a l , we condition" assumption p (77) i < f o r some i . social-welfare intuitive operator w i l l Section inequality c o r r e s p o n d s to the second its measurabi1ity (0) i with s t r i c t from the p n operators. Unanimity Amongst Principle, a the which says that p, = ... = p = p => T(p,,...,p n whatever sense p e n ( s e e Axiom A b e l o w ) . i f the s c a l e s intercomparable, the preferred of alternative difficulties mentioned of b e l i e f This condition u s e d by associated i n Weerahandi belief be with & Zidek "outside" s u c h as r e q u i r i n g ascribed only the d i f f e r e n t i . e . i f t h e r e e x i s t s an quantification ) = p n a comparing value of makes experts are standard that t h e most one. degrees of b e l i e f (1981); they are for analogous The are to 98 those which a r i s e i n the theory of u t i l i t y when one attempts compare " p r e f e r e n c e s . " taken to overcome Although v a r i o u s this problem q u e s t i o n remains l a r g e l y unsolved 1970). In the case of degrees in approaches the have been l a t t e r c o n t e x t , the ( c f . Luce & R a i f f a of b e l i e f , to 1957; the d i f f i c u l t i e s are r e f e r r e d to above d e r i v e from the e x i s t e n c e of Sen which possibilities which have not yet been i d e n t i f i e d and which, t h e r e f o r e , are not included in 0. Situations n a t u r a l to normalize p as p /p (6 ), i i d i s t i n g u i s h e d s t a t e i n 0. natural. dominating In B being some In o t h e r s , p /Sup{p (6)\6eQ} 0 i more are c o n c e i v a b l e where i t would be others 0 still, i i there might fixed and might be be a natural measure u on 0 with respect to which every p could i be normalized; this, of course, i s the very important case which we d i s c u s s e d i n Chapter particular choice seems 2. In general dictated. above-mentioned infinite division p 's i M~measure. Notwithstanding intercomparability time being t h a t preserving S e c t i o n 3.3, difficult by it pooling an these problems reasonable operators attempt problem total mass will involving to for be combining the certain when the 0 has with an the s h a l l assume f o r the use made no example, associated of s c a l e s of b e l i e f , we is though, Furthermore, a l t e r n a t i v e s would not always be f e a s i b l e as, f o r special at pooling (local) unanimity P-functions. solving of the In more incomparable 99 propensities. 3.2 A c l a s s o f l o c a l We apart now p r o p o s e c e r t a i n from l o c a l i t y , reasonable satisfy acts the for weak and a p p e a l i n g the quasi-arithmetic These proven in interesting feature of t h e theorem arithmetic means assumption of to the that a (defined context continuous relating characterization operators another o f smoothness a p p e a r s considerations requirements which, that operator pooling any should operator "axioms" a r e seen t o c h a r a c t e r i z e w e i g h t e d means are conditions r o l e of p o o l i n g now on, i t i s u n d e r s t o o d P-functions). was operators embrace t h e m i n i m a l candidate (from on which pooling by i s that in linear Aczel their n of the result An a l l quasi- variables, axioms. s c a l e s of b e l i e f and a (1948). although i n the l i s t the below), no When a r e added, a logarithmic pooling are obtained. Definition 3.2.1 n A transformation T:I1 —>IT i s c a l l e d operator i f f there exists function i//:(0,°°)—> R w i t h T(p,,...,p for some f i x e d w e i g h t s a quasi-arithmetic a c o n t i n u o u s and inverse iK 1 such strictly increasing that n ) = </r [ Z w <Mp )1 n i=1 i i n £ 0 with Z w = 1. n i= 1 i 1 pooling (3.2.1) 100 Important examples of are t h e l i n e a r o p i n i o n p o o l quasi-arithmetic pooling n Z w p i=1 i i [\p{x)=x], n w(i ) n p [i//(x)=log(x) ] i=1 i opinion pool pooling operator and operators the l o g a r i t h m i c t h e root-mean-power n c 1/c c ( Z w p ) [v//(x)=x ,c>0] which i n c l u d e s the i=1 i i first one as a s p e c i a l case (C=1) and t h e second as a l i m i t i n g case (c->0). The b a s i c p r o p e r t i e s of t h e q u 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 t h e case where the p 's a r e r e a l numbers. E s p e c i a l l y noteworthy i s the f a c t i that the function v// i s unique o n l y up t o an o r d e r - p r e s e r v i n g a f f i n e transformation ax+b, a > 0. T h i s r e s u l t we r e c o r d as Lemma 3.2.2 n L e t w,,...,w > 0 be f i x e d w i t h Z w = 1 and, f o r j=1,2, l e t n i=1 i n G (x) = }p~ [ Z w \p (x ) ] j j i=1 i j i 1 such that G (x)=G (x) 1 2 be two q u a s i - a r i t h m e t i c weighted means whenever x e l f o r a l l i=1,...,n, I i being some open i n t e r v a l i n (0,°°). strictly a>0. p o s i t i v e w ' s , then i i// 2 = I f there e x i s t a t l e a s t n a i ^ + b on I f o r some a , b two e R, 101 Proof; This result i s s t a t e d and i n Hardy, L i t t l e w o o d We could will now & P61ya present reasonably be (1934). four required these axioms c h a r a c t e r i z e p r o v e d as Theorem 83 page 66 • axioms which to s a t i s f y . the on pooling It w i l l quasi-arithmetic operator turn out pooling that operators 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 (1978) " p r i o r - t o - p r i o r coherence" "pooling be Since opinions the ultimate construct the In condition T (p 1 f order their all ...,p of sequential various example. of pooling be called a l l the manuscript, Zidek's stipulates i n any a combined it pooling Weerahandi that order." P-functions experts, i n which t h i s ) = T (T pooling, (p,,...,p 2 k-1 where i t s domain. coherence axiom functional can k k=2,...,n, dimension of & is P-function is plausible i s done s h o u l d 6 to Zidek be express as k for that beliefs actual immaterial. this the and done s e q u e n t i a l l y and objective something representing that can axiom Weerahandi this the This because point ) the will of i t may weights be best (3.2.2) k on T i t conveys the formulation r e l a t i o n s h i p amongst experts. subscript Although i s inadequate ),p k-1 the indicates basic idea the of prior-to-prior i n d u c e an ascribed illustrated undesired to with the an 102 Suppose t h a t expert we a r e i ' s opinion, dealing with p , has a weight n=3 w relative 1 / < 1 w + w 2 2 that o f w,+w . 2 w,+w _ other of experts 3 1 2 3 T (p,p ), 2 = w, b e i n g attributed t o expert 1 2 a n d 3, a n d n o t o n l y carry a i t follows 3 arbitrary example, that w i n (0,1). 2 = In determines the their the d i f f i c u l t y the"inner 2 as i t appears sum!! and 2 appears this problem, by i n thef i r s t i t will 2 totheir t h e source been p r o p e r l y slot be n e c e s s a r y keeping This components, o f an o p i n i o n to distinguish calibrated. i as of experts 1 T ." 2 To a v o i d a l l pooling weights which the underlying an (3.2.2) of expert of the "outer i sirrelevant from i nEquation opinion track of the s p e c i f i c various arose T " has the opinion i - t h c o m p o n e n t , i = 1 , 2 , whereas t h e j o i n t ascribe implies 1 a n d 2, p , s h o u l d = = ofw to i t s f i r s t f r o m w h i c h we c o n c l u d e 2 i n t h e use of T when k=3. T h e r e , operators is w 2 unacceptable. t h e above ambiguity a weight to p 3 w = w /(w,+w ), 3 respect 3 has a weight o f w , which T (p,,p ,p ) of both e x p e r t s In that Since words, t h e w e i g h t clearly its opinion and i n turn w weights is = 2 j/w7 w, p, with 2 However, e x p e r t t h e combined weight of opinion 1 Then 3 i s o T ( p , p ) must a s s i g n 2 ) r component. that weight and t h a t o f w >0, w,+w +w =1. i the experts idea they being as long a s i t has 103 Change in notation When a p p r o p r i a t e , we shall T (pi,...,P I write n denote the j o i n t functions, T G i s local, (p^G),...,? n In we shall ) to n propensity and amalgamated u s i n g weights w £0, i=1,...,n, i n T n of n experts p , a r e t o be w e i g h t e d i and c e r t a i n If opinion n E,,...,E whose n write formula n Z w =1. i=1 i T (p,,...,p | n n )(d) = n (0)|w,,...,w ) . n n adopting this convention, i t i s understood that T n should T satisfy (p,,...,p n n |w,,...,w ) n T n where T i s any speaking, list this of axioms. property (p,, n (p ,...,p |w ,...,w ) T(1) T(1)T(1) T(1) permutation of requirement However, the s e t {l,...,n}. could be i n c l u d e d we p r e f e r t o r e g a r d of a l l P-function Axiom A ( U n a n i m i t y T = pooling (3.2.3) Strictly i n the f o l l o w i n g i t a s an i n t r i n s i c operators. Principle) ,p |w,,...,w )= p whenever p , = ... = p = p n n n 104 Axiom B ( P r e s e r v a t i o n o f t h e o r d e r i n g T (p 1 f ...,p n | w )(6) w n strict a POB p o o l i n g (p,, where p = T (p k an a r b i t r a r y 1 f ,w ) n the inequality q (0) j i s equivalent being f o r some 1 ^ j ^ n w i t h t o saying that T n coherence) T ( p , p ,...,p |w,w ,...,w ) n-k+1 k+1 n k+1 n k . . . , p |w,/w,...,w /w) a n d w = I w ( i f w=0, p k k i=1 i = P-function). Axiom D ( M o n o t o n i c i t y T —,w )(0) n operator. ,p |w,, n n If this < 1 f a l l 1<k<n, For is p (d) j When T i s l o c a l , n Axiom C ( P r i o r - t o - p r i o r T f o r a l l 1<i£n, i f , i n addition, 0<w < 1 . j is < T (q,,—,q |w n n n whenever p (d) £ q (0) i i of b e l i e f s ) of weights) w <w*, w = 1-w +w* a n d t h e r e i i i i ( p n p n | w, , — , w ) ( 0 ) n T provided that < ( n e x i s t s j * i s u c h t h a t w >0, t h e n j p p n p (6) = max{p ( 0 ) | l < k < n i k |w,/w,...,w*/w,—,w i n & w >0}. k /w)(6) 105 When n>4, Theorem Let 3.2.3 T we have t h e (Aczel be l o c a l following 1948) and s u p p o s e t h a t there exist T ,...,T that such 2 n n-1 Axioms A-D above be satisfied. Then T is a quasi- n arithmetic Remark pooling operator. 3.2.4: The e s s e n c e of the proof However, we obvious redundancies. adaptation Proof have First, adapted we (1948) h i s a x i o m s and f r e e d For of h i s p r o o f o f Theorem i s contained in Aczel's completeness, them we some necessary i s g i v e n below. 3.2.3: show t h a t define from the T ,...,T are l o c a l 2 whenever n-1 that, paper. n-2 T i s . For n k :(0,») — > ( 0 , » ) , k=2,...,n-1 functions G by k letting (yi,...,y G k |v,,...,v k ) = k T (p,,...,p n for any p 1 f ...,p G 's k are k ell w i t h p k well-defined. ,...,p (0)=y i |v,,...,v k ,1<i^n. k Since T i Furthermore, ,O,...,O)(0) is local, the n we c a n use prior-to-prior 106 coherence t o see that T ( p i , . . . , 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) n-k+1 k k k v=v k + 0 + . . . + 0 = v . k k i s local by unanimity G (p,(0),...,p ( 0 ) | v . . . , v ) k k k 1 f f o r a l l k=2,...,n. Next, we d e f i n e a f u n c t i o n X : [ 0 , 1 ] — > [ a , b ] and v e r i f y that G ( x ( s ) ,x(t) | 1-w,w) = x[d-w)s+wt] (3.2.4) 2 for and Therefore, T (p,,...,p |v,,...,v )(0) = k k k always, i . e . T = p k a l l s,t,w € [0,1 ]. In f a c t , as i s shown below, we may l e t x(«) = G (a,b|1-w,w) f o r a l l 0<W<1, so that x(0)=a and x(1)=b by 2 unanimity and p r i o r - t o - p r i o r coherence. Using Axioms A and B, we see that a = G (a,a|1-w,w) < x(w) < G (b,b|1-w,w) = b f o r a l l 2 w e (0,1), increasing and 2 i t follows from Axiom with Axioms D that x i s s t r i c t l y in (0,1). Conjugating Equation (3.2.3) A and C, we have successively G (x(s),x(t)|1-w,w) 2 = G [G (a,b|1-s,s),G (a,b|1-t,t)|1-w,w] 2 2 2 = G [a,b,G (a,b|1-t,t)|(1-w)(1-s),(1-w)s,w3 3 2 = G j a , b , a , b | ( l - w ) ( l - s ) , ( l-w.^s, ( l - t ) w , t w ] = G«[a,a,b,b|(1-w)(1-s),(1-t)w,(1-w)s,tw) = G [a,b,b|1-s+ws-wt,(1-w)s,tw3 3 = G [a,b|1-s+ws-wt,s-ws+wt] 2 107 = x(s-ws+wt) = x f ( 1 - w ) s + w t ] , so t h a t The it (3.2.4) h o l d s t r u e . key o b s e r v a t i o n i s t h a t x i s c o n t i n u o u s i s surjective hence it has on [ 0 , 1 ] . on [ a , b ] by t h e I n t e r m e d i a t e an i n v e r s e x rewrite Equation _ 1 Value :[a,b]—>[0,1]. F o r then Theorem T h i s a l l o w s us t o (3.2.4) a s G (y,z|1-w,w) = [ ( 1 - w ) x " ( y ) + w ~ ( z ) ] 1 2 X where y and z a r e i n [ a , b ] a n d 0^w<1. We argue point For of d i s c o n t i n u i t y ye[a,b] such a (x~ will 1 be t h e f u n c t i o n (3.2.1).) f o r x's c o n t i n u i t y 3 (3.2.5) 1 x \p o f E q u a t i o n and V by c o n t r a d i c t i o n . Suppose t o f x, s a y t o t h e r i g h t . 0 is a Then s>0(s+t e[0,1]=>x(t )<y<x(t + s)). o 0 0 number y e [ a , b ] , w r i t e y = G [X(t),y|1/2,1/2] 2 for t all te[0,l]. By Axiom B, we G [x(t),x(t )|1/2,1/2] 2 < 0 have G [x(t),y|1/2,1/2] = y < G [ (t),x(t +s)|1/2,1/2], t 2 2 i.e., using Equation X 0 (3.2.4), x[(t+t )/2] < y < 0 xt(t+t +s)/2] 0 t for a l l s>0 w i t h (t+t +s)/2 0 e [0,1]. T h i s w o u l d show t h a t x i s d i s c o n t i n u o u s a t a l l ( t + t ) / 2 0 h o w e v e r , a monotone f u n c t i o n n e v e r number Hence, Using of discontinuities x i s continuous has more (Theorem 4.30; R u d i n e v e r y w h e r e and E q u a t i o n i n d u c t i o n , we w i l l now p r o v e G ( y i , . l . , y |v ...,v ) k k k 1 f than a (3.2.5) o b t a i n s . k [ Z v x " ( y )] i=1 i i 1 x countable 1976, p. 9 6 ) . that = e [0,1]; 108 k for a l l 2^k<n and y i c [a,b], deduce C that G from Axiom v £0, i 1. Z v = i=1 i (yi,.««,Y k+1 Indeed, |v ..,v ) 1f k+1 we equals k+1 k i f v = I v = 0 and G [G (yi,...,Y Ivt/v,...^ / v ) , y |v, y v 2 i=1 i k+1 ] k k k k+1 otherwise. k+1 However, we know by hypothesis that G k and so G (yi,...,y |v,/v,...,v k (yi,...,y k+1 k /v) = xt Z (v / v ) - x " ( y )] k i=1 i i . 1 | i,...,v v k+1 G [x{ 2 Using Equation k+1 ) equals k Z (v / v ) . x * ( y )},y |v,v ] . i=1 i i k+1 k+1 1 (3.2.5) now, we find that this last quantity equals x[v. " {x( 1 X k Z (v / v ) . x ( y ))} i=1 i i _ 1 + v -x (y _ 1 k+1 )], k+1 k+1 which i s nothing but xt Z v x " ( y ) ] . i =1 i i 1 To complete the p r o o f , i t remains t o show t h a t i f yi,...,y a r e n any i n (0,»), strictly then ^:(0,»)—>R exists which i s continuous, i n c r e a s i n g , and such that n G (yi,...,y |w ...,w ) =tf-'lZ w t//(y )] n n n i=1 i i 1f n with w,,...,w ^0, Z w = 1 . n i=1 i (3.2.6) 109 For this, we c o n s i d e r t h e n e s t e d sequence o f c l o s e d i n t e r v a l s I m = [1/m+l,m+1] a=l/m+1 i n (0,°°); we c a n r e p e a t a n d b=m+1 t o p r o v e strictly increasing Equation the existence function x" m for yi,...,y (3.2.6) h o l d s Since I, n \p~ [ Z w \jj (y ) ] m i=1 i m i yfr* = m - ]/a m m for some a=1,b=0). a,b e R from m Theorem the proof m+1 positive. = Define o f ^^ on I , m>2; we m , s i n c e \p* = a^* + m+1 m b a>0, and \p* = i//* = i / ^ on I , ( i . e . m+1 m o f Theorem 3.2.3. • 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 operator t o be l o c a l , beliefs, as w e l l as t o pool to I m with a quasi-arithmetic pool. then 1 We l e t \p = ^* on I , m=1,2... m m T h i s completes axioms I and Lemma 3.2.2 t h a t ii m from s o t h a t ^* i s an e x t e n s i o n m have t h a t ^* extends m+1 continuous n [ Z w V/,(y ) ] = 1 i=1 i i i n I and m e R with a s t r i c t l y m f o r some a ,b mm b m a ^ :I — > [ 0 , 1 ] such t h a t mm e I and $ i n s t e a d o f \//, m m on I , , we c o n c l u d e 1 a \//, + b m m i s contained of with = 1 n m=1,2... t h e argument a b o v e If a decision t o preserve satisfy of p r i o r - t o - p r i o r he must u s e F o r m u l a coherence (3.2.1), h i s experts' P-functions. f o r using maker wants h i s unanimity the support and t h e o r d e r i n g of eminently reasonable and m o n o t o n i c i t y of weights, with two pooling some u n d e t e r m i n e d The r e q u i r e m e n t i//, t o t h a t n£4 i s not 110 really and limiting, assign hypothesis, b e c a u s e one c a n a l w a y s throw them a z e r o therefore, intercomparable; Principle conditions reflect candidate to Clearly, there of P - f u n c t i o n is no level unique to long say as that that pooling this Aczel's any operator serious would be i n the problem of below). with a l l choices In other pooling invariance in the problem, reasonable to restrict operators to those with respect used in statistical the may class d e c i s i o n theory presently, Bergson-Samuelson i t will best speaking, goal. -as In c e r t a i n this will 3.2.9 symmetry a result- i t could quasi-arithmetic T h i s method or seem pooling or i n v a r i a n t is frequently a decision rule r u l e does n o t e x i s t . prove the considerations some f o r choosing the on ( c f . Theorem which a r e symmetric to c e r t a i n operations. question, at depending special induce of (3.2.1)? be m e r i t o r i o u s ; P-values and operators where an o v e r a l l vary however, t o the s c a l e s of b e l i e f this Generally the intended o f \(/ w i l l cases, Formula way t o answer of a b s t r a c t i o n . a t hand t o g e t h e r circumstances, see As requirements f o r s e l e c t i n g a "good" <// w i l l application cases 3.1). fair are f o r the Unanimity one c h o o s e t h e f u n c t i o n $ i n not a t t h i s criterion in seems the minimal role s c a l e s of b e l i e f t o meet. should relating the various e s s e n t i a l i n order i t opinions The o n l y p o s s i b l y c o n t r o v e r s i a l (cf. Section valid, the How happen is t o make s e n s e is least i s that this assumption expected weight. i n dummy As we shall s u c c e s s f u l when s e a r c h i n g fora s o c i a l - w e l f a r e f u n c t i o n , amongst others. 111 Examples 3.1.1 & 3.1.4 In this acceptable (continued) example, joint we utility are concerned function, with finding W(u ...,u an ), which might 1f n well be a B e r g s o n - S a m u e l s o n purpose, u we social-welfare function. make t h e somewhat r e s t r i c t i v e 's a r e c a r d i n a l utility functions For assumptions w h i c h a r e bounded that that the from below The first i and, more i m p o r t a n t l y , condition and implies which that are u +c > 0 f o r some ceR i n d e p e n d e n t i so t h e u 's c a n be s i m p l y i The second condition preserves unanimity. subject intercomparable. treated makes i t as propensity possible to t o some i n t e r p r e t a t i o n . For instance, functions, require However, t h i s c o m p a r a b i l i t y of i that assumption W is t h e u 's m i g h t be i taken to be either scale 1970, p. 106), depending that the comparable transformed comparable utility functions, f o r a l l a > 0 and b e R or only to that postulates be to postulate v = au + b , are i i i when b = b for a l l i respectively. Given four c o m p a r a b l e (Sen on whether we a r e w i l l i n g i 1<i<n, or f u l l y entirely the u t i l i t i e s are f u l l y (Axioms A-D) a r e e a s i l y compatible behaviour with of a our concerning the function. By Theorem 3.2.3, t h e j o i n t comparable, say, Aczel's interpretable intuitive "reasonable" utility and a p p e a r expectations social-welfare u = W(u ...,u ) 1f n 1 12 should t h u s be given by n u ir [ Z w yp(u ) ] = 1 i=1 for R some c o n t i n u o u s and and w,,...,w n natural £ strictly our a l l a>0 and beR increasing function pooling w(au,+b,...,au +b) n for i n I w =1. i =1 i 0, t o demand t h a t i = Furthermore, operator W a«W(u,,...,u ^:(0,»)—> i t would seem obeys ) + b (3.2.7) n such that au +b > 0 on the whole domain of i u ,i=1,...,n. This invariance property of W guarantees that no i dilemma and will transforming As imply it a precise linear operator from the turns Proposition The arise interchanging s c a l e s of out, form for this the operations of pooling belief. extra requirement is sufficient to w. 3.2.5 opinion which pool is satisfies the only Equation quasi-arithmetic pooling (3.2.7) f o r a l l a and b in (0,»). Proof: It i s well p. 68) known that the (Theorem 84; only Hardy, L i t t l e w o o d quasi-arithmetic & P61ya means M ( x . . . , x 1934, ) which 1 r n 113 satisfy M(ax,,...,ax ) = a«M(x ...,x are ) for a l l x ...,x 1 r n , 1 f n t h e w e i g h t e d means o f o r d e r a , n [ which we have those, only already / a ] , a*0; i w (i ) I I x i =1 i , a=0, encountered M, o b e y s t h e 1 a w ( i ) x I i=1 n M (x , . . .,x ) = a 1 n second i n Chapter condition 2 f o r 0<a<1. M (x,+b,...,x a (x,,...,x n In the )+b f o r a l l x,,...,x n ,b > 0. n language W(u,,...,u Of +b) = a M a>0 n ) = n of pooling operators, n I w u f o r some w £ 0 w i t h i=1 i i i is quasi-arithmetic and C o n d i t i o n A similar could argument this means n Z w =1 i=1 i (3.2.7) h o l d s . that whenever W • a l s o be u s e d i f t h e u 's were only i assumed t o be s c a l e c o m p a r a b l e . a d d e d bonus that some b = the satisfy case, W w o u l d have t h e to satisfy W(au,+b for In t h a t 1f ,au +b ) = a « W ( u n n n I w b , b > 0. i=1 i i only the dictatorships; —,u ) + b n I t may be o f some i n t e r e s t quasi-arithmetic following 1 f pooling generalization operators, of t o note W, (3.2.7) which are 1 14 W(a,u,+b,,...,a u +b ) = n n n W(a ...,a )«W(u ...,u 1 r If we t h i n k o f a " p r o p e r " takes account of the individuals concerned, interpreted as saying interpersonal Example 3.1.2 Suppose ) + W(b,,...,b 1 f n n s o c i a l - w e l f a r e f u n c t i o n as preference then that patterns this such last of one each observation a function will comparisons of u t i l i t y ). n which of the could not e x i s t be unless are possible. (continued) that p 1 f ...,p e II r e p r e s e n t the n o p i n i o n s of a n group of e x p e r t s and t h a t e U are their respective n (subjective) {X,,...,X }, likelihoods has been f o r 0 given observed that a single data-set, D = by a l l t h e e x p e r t s . We shall s assume t h a t upon o b s e r v i n g accordance with D, e a c h e x p e r t the f o l l o w i n g q i s the o p i n i o n rule: $ «p , i = 1 , . . . , n , i i i where q updates h i s b e l i e f s i n of the i - t h expert (3.2.8) given D. This i formula the i s i m p l i e d by B a y e s ' Theorem normalization constant i n the case i s irrelevant of P - f u n c t i o n s : as p r o p e n s i t y functions 1 15 are to be treated -and sometimes c o n s i d e r a t i o n o f - b e t t i n g odds. exist is i fp i s sufficiently i regarded as a l a b e l m u l t i p l e s of p The i i n t e r p r e t e d through M o r e o v e r , t h i s c o n s t a n t may n o t improper. So i t i s o m i t t e d and q i f o r the equivalence c l a s s of a l l constant (Novick natural even & Hall 1965, pp. 1 1 0 5 - 1 1 0 6 ) . c o u n t e r p a r t o f 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 here: s h o u l d P - f u n c t i o n s be c o m b i n e d b e f o r e or a f t e r observation of that i f i t i s decided t o pool f i r s t , functions joint the of the sample evidence D. the discrepancy experts will P - f u n c t i o n c a n be u p d a t e d . Note between the the likelihood h a v e t o be r e s o l v e d b e f o r e t h e The i d e a , h e r e , i s to pool 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 o f o p i n i o n ( P - f u n c t i o n s ) f r o m t h e same e x p e r t s , i t w o u l d use the same p o o l i n g formula seem as f o r the p r i o r s . w o u l d 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 o f p o o l i n g commute. An o p e r a t o r w h i c h d o e s t h i s w i l l p o s t e r i o r coherent" as this a substitute thesis, a f t e r Weerahandi & Zidek f o r Madansky's term natural to Moreover, i t and be c a l l e d updating "prior-to- ( 1 9 7 8 ) who u s e d i t "external Bayesianity;" in t h e two e x p r e s s i o n s a r e now v e s t e d with different meanings. Definition 3.2.6 n We say that 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 coherent i f f T ( * , p , , . . . ,* p ) = T ( $ , . . . , $ )«T(p,,...,p ) n n n n 1 (3.2.9) 116 for all * ,p i Note that Assuming this that independent theory e II, i = 1,...,n. i at definition holds, (3.2.8) criterion least, our does involve locality. p r i o r - t o - p o s t e r i o r coherence for selecting search not a pooling operator; f o r such operators could i s an thus, extend in to n all T:I1 —>n. applications suggest that Aczel's four operators. formulas postulates, In that singles (3.2.9) will pooling However, case, out the arguments given be and satisfy quasi-arithmetic pooling should i . e . be it is easy logarithmic local to see opinion that pool. above Property In fact, we show more: Proposition 3.2.7 n Let T:II —>n ordering There be of exist a local beliefs pooling (POB) and operator )(e) = n n preserves the i s p r i o r - t o - p o s t e r i o r coherent. w(1),...,w(n) > 0 such n T(p,,...,p which that w ( i) [p i= 1 (3.2.10) (e)] i n for a l l 0e6 and (p,,...,p ) e U , i.e. T is a logarithmic n opinion pool. Moreover, n Z w(i) i=1 unanimity. = 1 whenever T preserves 1 17 Proof: Write T(p,,...,p )(0) = G(p,(0),...,p (0)) n n e P i , . . . , P Using n . for a l l 0e0 E q u a t i o n ( 3 . 2 . 9 ) , we s e e t h a t and G(x»y) = n n G(x)»G(y) for a l l x Definition and 3.1.7 t h a t variables. ) = n to illustrate has c o l l e c t e d i t 2.1.3, we f o r some w ( i ) > 0 , follows conclude i = 1,...,n. that result, data suppose x,,...,x and that that from i n each of i t s n that The sum G(x,...,x)=x the Unanimity P r i n c i p l e . this some and increasing u n l e s s we r e q u i r e x>0, i . e . T s a t i s f i e s To (0,») , Lemma n w( i ) II x i =1 i n Z w(i) i s a r b i t r a r y , i=1 all in G is strictly Appealing G(x,,...,x y • a decision he for maker regards the n likelihood an function individual expert opinion. $ might = n=1 return this result would, Formula (3.2.10) o r p e r h a p s Both (3.2.11) the representative i f t h e sample likelihood. each of these items p r o v i d e s as n 1/n n [ $ (0)] i =1 i t h e n be u s e d t o o b t a i n Alternatively, just $ (0) w h i c h i likelihood, i s t a k e n a s a whole, (3.2.10) and likelihood in a possibly i n general, differ be would then r e n o r m a l i z e d form. This from t h a t (3.2.11) there w i l l of Equation (3.2.11). 118 The point, however, indicating, should be pooling in any c o m b i n e d , but such functions obtained likelihood distribution result will which would just Law the of the be obtained exp{l(f ,f 0 discrimination this differs of the measure, * i f 0 i s the "true realization of the highly data are the of in the paragraph their joint usual sampling way, the n 1 /n [ II 4> ] i=1 i quantity (3.2.11). joint The latter i f n—>», the converges to a is Strong constant Kullback-Leibler O (0) = f(x i density and )}, where I d e n o t e s t h e 0 means last the from the it not functions a i n the f o r m e r and, that is selected. computing from E q u a t i o n rescaled version of providing measurements inverting L a r g e Numbers i m p l i e s multiple is presented first This operator which p r o p e n s i t y it data by n II * . i=1 i be context, independent then pooling once t h e y have been found and the rather if from is that given Incidentally, are is |0), f(«|0) i s the sampling i s t a t e of nature," random v a r i a b l e 0. dependent, say and the true other hand, x,, i = 2 , . . . , n , then I f , on x = 0 O is the i Equation (3.2.11) likelihood In finding Let us would give, or very nearly give, the joint itself. Section 3.3 non-local mention w e i g h t e d mean o f below, we shall address p r i o r - t o - p o s t e r i o r coherent in order passing a (M that a defined a the pooling problem operators. characterization above) of obtains of if the the 119 validity of Condition t h e r e was m u t u a l • •• ^ (3.2.9) i s l i m i t e d agreement a p r i o r i on t o t h o s e c a s e s where the likelihood 4> = • n Proposition 3.2.8 n Let T:I1 —>n be a quasi-arithmetic pooling operator. If T satisfies T($p,,...,$p ) = ••T(p,,...,p n for e n, a l l $, p,,...,p then T = M n Proof: This ) n f o r some a e ( 0 , » ) . a i s because the weighted means o f o r d e r a>0 are the n only quasi-arithmetic "homogeneous." P61ya (1934). Example 3.1.3 In this means M:(0,») —>(0,») which are See Theorem 84 on page 68 i n H a r d y , L i t t l e w o o d & • (continued) case, the pooling operator P(L 1 f ...,L ) is n automatically local, interpretation of A c z e l ' s axioms causes For set instance, set corresponding 0 = {H } 0 i s a singleton. no difficulty S 2 s h o u l d be more s i g n i f i c a n t , if the P-values P-values in S . 2 in As The either. Axiom B e x p r e s s e s t h e e v i d e n t r e q u i r e m e n t S, o f P - v a l u e s another because a s a whole, that a than S, a r e s m a l l e r t h a n t h e another example, the 120 inequality min{L |i=1,...,n} < P ( L , , . . . , L i (a n c o n s e q u e n c e o f Axioms A and the fact that ) ^ max{L because B taken |i=1,...,n} i together) accounts t h e d a t a upon w h i c h J , , . . . , J are for based n cannot or be combined incomparable qualitative (either because they are u n a v a i l a b l e due to differences aspects of t h e v a r i o u s d e s i g n s ) , we the combination than directly t h e most test in t o g i v e us more (least) optimistic the (less) quantitative do not expect confidence of the o b s e r v e d levels or in H 0 L . i Theorem 3.2.3 suggests that we use the test statistic n P ( L , , . . . ,L ) = n \p' 1 [ Z w ^ ( L ) ], i=1 i i where i//:(0,°»)—>R would be c o n t i n u o u s and its domain. strictly statistic Liptak Moreover, increasing S 1958, will p. with suitable statistics was continuous produce 176), P[\p] using fact that increasing in general transformation same o n e - s i d e d first can x(s) test r e s t r i c t our a t t e n t i o n n ( L , , . . . ,L ) = Z w v//(L ) n i=1 i i weights we the the strictly w £0, i n Z w =1, i= 1 i say. i n t r o d u c e d by L i p t a k any of as a S (cf. to (3.2.12) This family (1958) on and of test comprises 121 ( i ) Good's ( 1 9 5 5 ) weighted v e r s i o n of F i s h e r ' s omnibus procedure [\/>(x) = l o g ( x ) ]; U" 1 (ii) the so-called i n v e r s e normal procedure (x) = JI (-»,x)-exp{ (-1/2) (y +log(27r) }dy]; and 2 recent logit statistic l o g ( x / ( 1-x)) ]. r e s t r i c t e d to different of In the two ( 0 , 1 ) , but Mudholkar l a t t e r cases, t h i s can form of Theorem 3.2.3 take v a l u e s i n The & be George the where the result (Liptak Each member P[^] against unbiased t e s t the of i// i s a p p e a l i n g to a P - f u n c t i o n s would only vindicates the a r i t h m e t i c weighted means i n t h i s p a r t i c u l a r test ( 1 9 7 9 ) [^(x)= domain j u s t i f i e d by more [0,1]. following Theorem 3.2.9 ( i i i ) the of the quasi- application. 1958) of the class (3.2.12) yields a some s p e c i f i c a l t e r n a t i v e . f o r the o r i g i n a l test use sample c o n s i s t i n g statistics J 1 r ...,J most powerful Moreover, P[\p] of the is an P-values whenever are. n Proof: A d e t a i l e d proof of t h i s theorem i s c o n t a i n e d i n L i p t a k ' s paper. we However, statement is statistic P[i£] "condition 1." a direct of the not to consequence mention of the that the f a c t that any first test form ( 3 . 2 . 1 2 ) s a t i s f i e s Birnbaum's ( 1 9 5 4 ) alternative i s a d m i s s i b l e may constitute like • In g e n e r a l , the P[i//] would a be strong a g a i n s t which a p a r t i c u l a r q u i t e obscure, so this b a s i s f o r choosing one result test does form of P[<//] 122 over a n o t h e r . only Moreover, the vaguely. difficulty In h i s p a p e r , L i p t a k by \p using distribution normal for the procedure." particular He point of possible distributions to l e t the taking t e s t i n g problems, family. value of of nonparametric Another Roy's way sizes required normal from a of i n c l u d i n g t h o s e where the by large d e n s i t i e s belonging Scholz (1981) "proper" his to proposed function v/> by largest proposal (1953) u n i o n - i n t e r s e c t i o n to any attain & Folks least by the two P[^]'s as given as an p r i n c i p l e to w e l l - k n o w n c o m p e t i n g methods, procedure. In the John P e t k a u for bringing attention): following fact, a a this Fisher's Bahadur stronger (we are compute ratio statistics including consequence to significance that i n the be limiting test small showed efficient, would i . e . the equally (1971) a u t h o r s has our this class describes (1967) e f f i c i e n c y , (3.2.12) Thus, L i t t e l l He of c o m p a r i n g sample at that setting. Bahadur other generated "inverse convenient for a More r e c e n t l y , P[<//]. their always optimal i . e . the proof- being this cumulative 4> ( s u i t a b l y s t a n d a r d i z e d ) w h i c h y i e l d s t h e application form -without from are the normal N(0,1), claimed specified to circumvent of P-values themselves choose the that possible proposed inverse v i e w , was hypothesis exponential the apart one-sided the = standard choice, computational alternative is generally of the levels. method s e n s e , as Liptak's result of of is three inverse the same g r a t e f u l t o Dr. s e c o n d p a p e r of L i t t e l l & Folks A. to 123 3.2.10 Theorem Let (Littell P(L,,...,L ) be 1973) & Folks any test statistic which preserves the n ordering of hypothesis beliefs (Axiom Bahadur sense, Proof; If (POB) B). c o n c e r n i n g the v a l i d i t y Then P i s a t most as as F i s h e r ' s P(L,,...,L ) of t h e efficient, omnibus procedure. satisfies Axiom B, then null in the the statistic n S(J,,...,J ) = -P(L,,...,L n for large values which appears Remark It is ) which rejects the null hypothesis n on of page S satisfies 193 the c o n d i t i o n in L i t t e l l of the (1973). & Folks theorem • 3.2.11 not too increasing difficult to find i//:(0,1)—>R functions continuous and strictly f o r which the test statistic n I w(i)i£(L ) w i l l i =1 i have the same Bahadur efficiency n corresponding one could take distribution variable. a weighted ^ Fisher to be o f a gamma, an However, Theorem statistic of the form procedure the of . the For example, cumulative i n v e r s e - G a u s s i a n or a L a p l a c e 3.2.10 asserts (3.2.12) will the w( i ) n L i =1 i inverse as that no test s u r p a s s the random based on omnibus procedure. In summary, a s e t o f developed which was weak and appealing shown t o c h a r a c t e r i z e the conditions was quasi-arithmetic 124 pooling the operators scales of express t h e i r operators (3.2.1). belief preserving. be A the the s c a l e s of legitimately locality application, acceptable belief assumption out look "optimal" at the Deriving It as opinion pooling ordering which of problem. are not to operator When saw that, reduce must be beliefs. solution At approach to to the end depending class in we to of imposing rule take a P-functions comparable. the so-called a reasonable is finite, the for pool that is made sometimes problem of p o o l i n g necessarily pooling unanimity the may to pertaining taken even when d e g r e e s of © the following section, opinion L, also that experts be was reductions section pool, that to of pooling choice belief another product i s p r i o r - t o - p o s t e r i o r coherent drawn between our (1950) the logarithmic intercomparable. derived In challenging belief the imposed consequent so conditions However, c a r e the more invariance be i s shown i n t h i s logarithmic function can solutions. whose s c a l e s of 3.3 extra various supposed F u r t h e r m o r e , we operators. such c o n d i t i o n s , the were i n t e r c o m p a r a b l e , mathematical convenience. on assumed, f u n d a m e n t a l l y , w h i c h were u s e d by opinions could I t was the and of a are formula preserves section, a parallel P-func.t.iojns multi-person general and cooperative Pnot is the is Nash's decision 125 Definition The 3.3.1 general logarithmic L(p for 1 f ...,p a l l p,,...,p n ) opinion i s d e f i n e d by n w (i ) C ( p , , . . . , p )• n [ p ( 0 ) ] n i=1 i = n e pool and 0e0, where (3.3.1) n C:I1 — > ( 0 , » ) is some n unspecified constants The operator and w(1),...,w(n) are non-negative n Z w ( i ) > 0. i =1 such t h a t function choices function L defined C depends on ( p of p . . . , p 1 f above . . . , p ). n is not If C(p local 1 r because ...,p ) = 1 for a l l n e II, t h e n L r e d u c e s t o t h e l o g a r i t h m i c 1 f the pool n (3.2.10). the In a n t i c i p a t i o n of the developments below, make following Definition 3.3.2 n The we relative propensity w h i c h maps any vector (n+2)-tuple (p i s a f u n c t i o n RP:II 1 f . . . , p ,0,TJ) n in n x© —>(0,») 2 n II x © 2 to the of quotients RP(p,,...,p ,0,7?) n It mapping is equivalence immediate relation = (p, ( 0 ) / p , (TJ) , . . . ,p ( 0 ) / p ( T J ) ) . n n that the n on D = II x 0 . 2 application I f two RP elements induces of an n II x © , 2 126 CIT say and RP(d ), d , are 2 t h e n D may 2 called be RP-equivalent decomposed into whenever RP(d,) = RP-equivalence classes n obtained through RP's regarded as a l a b e l shown, the logarithmic Definition i n v e r s e mapping. The s e t (0,») s e t f o r t h e q u o t i e n t s p a c e D/RP. following opinion property characterizes may be As w i l l be the general pool. 3.3.3 n We say that o p e r a t o r T:II a pooling c o n s i s t e n t (RP-C) i f f T ( p , , . . . , p ){6) n T(p!,...,p ) ( X ) n whenever RP(p,,...,p 1 f ...,q i s relative propensity )(TJ) T(q ,...,q 1 > n T(q,,...,q (3.3.2) ) U) n ,0,X) > n q —>n being a r b i t r a r y RP(q,,...,q ,7?,£), n Pi,--«»P , n o f n and 0 , T ? , X , £ belonging to elements n 0. To interpret Condition this (3.3.2) i n t o (i) T ( p p new concept, two p a r t s , )(c9) n > )(X) T(p,,...,p n i t i s useful namely T ( p , , . . . , p )(r?) n T(p,,...,p ) ( I ) n whenever R P ( p . . . , p ,0,X) 1 f > RP(p ...,p and T(p,,...,p )(d) n T(p,,...,p n )(X) T(q,,...,q = ,TJ,^); 1 r n (ii) t o decompose ){6) n T(q!,...,q ) ( X ) n n 127 whenever R P ( p , , . . . , p ,0,X) = R P ( q , , . . . , q n It are i s easy t o check equivalent that a good pooling c o n s e n s u s o f t h e form versus X are particular, requirement w i l l the the "consistency" again, time way than that any of D e f i n i t i o n Condition ( i i ) is as the C o n d i t i o n result a r e now of t h i s Theorem 3.3.4 Suppose that general preserve for TJ which preserve any says prior of the occurrence odds operator conditions versus the o r d e r i n g of 0 £." satisfies In this of b e l i e f s , 3.1.7, a n d -by way o f c o n s e q u e n c e i n Lemma 3.1.6. a s i m p l i f y i n g assumption; I t may only Here this be i n t e r p r e t e d i n t h e same ( i i ) which appears in Section i n a p o s i t i o n t o s t a t e and prove 3.1. the principal section. © contains logarithmic propensity should c o n d i t i o n which appears i t involves odds-ratios. We the ( i i ) together of these " t h e odds i n f a v o u r automatically in ( i ) and The f i r s t procedure better note sense that Conditions t o (3.3.2). ,0,X). n at least opinion consistent pooling three pool, L, distinct elements. is only the The relative operator. Proof: If T is any hypotheses of the immediate P-function pooling theorem, Condition implication that the operator (3.3.2) function satisfying the above the has Q(pi,...,p ,#,»?) n = 128 T(p 1 f ...,p n , 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 c l a s s e s o f D. (0,») Therefore, there exists s u c h t h a t Q=HnRP, t h e s y m b o l a mapping H : ( 0 , » ) — > n r e p r e s e n t i n g as before t h e composition of functions. Q — > ( 0 , » ) RP / • ( 0 , » ) Pick 0,7?,X, arbitrary that / / / / / 7» H / n three d i s t i n c t vectors i n / elements n (0,°°) . o f 6 , a n d l e t x a n d y be two I f p,,...,p Q(pi,...,P H ( p , ( 0 ) / p , ( i j ) , . . . , p ( 0 ) / p (T?)) n n n > 8 ,\) «Q(pi,... ,p , X , T ? ) n condition variables, that H so > 0 n w (i) n x i=1 i by an a p p l i c a t i o n i t s r o l e as a r e g u l a r i t y Consequently, Q(pi,...,p H ( p , ( 0 ) , ... ,p n with follows i n each some fixed = ( 0 ) ) « from the of i t s n numbers o f Lemma 2 . 1 . 3 ( h e r e , POB condition). ,0,T?) = n Q(p,,...,p , 0 , T ? ) n I t also i s non-decreasing so that H(x) = w(1),...,w(n) = = H( 1 / p , (T?) , ... , 1/p (T?)) = H ( 5 f ) - H ( y ) . n plays a r e chosen p ( 0 ) = x , p (T?) = 1/y a n d p ( X ) = 1 f o r a l l 1£i£n, t h e n i i i i i H(x-y) = RP-C n e n n w( i ) II [ p ( 0 ) / p ( T J ) ] i=1 i i for a l l 1 29 p 1 f ...,p e IT and 0,TJ e G, i . e . we have shown t h a t )(0) T(p,,...,p the function n C(p,,...,p = n is n i n d e p e n d e n t of 0! This theorem • is not e l e m e n t s , as t h e f o l l o w i n g Example Let n w (i ) n [p ( 0 ) ] i=1 i )(8)/ true if 0 contains counter-example exactly two shows. 3.3.5 n T:II —>n 0 = {0,r?} and d e f i n e by T ( p , , ,p ) = (p,+p )/2. 2 n Then T is relative preserving, As and propensity but c l e a r l y before, T * the problem i s not a d d r e s s e d h e r e . Equation (3.3.1) requirement Unanimity C(p,...,p) Principle intercomparable). role this If insist we is function that = also Note also that the e n experts' o r even the except t h e moment, i t i s n o t unanimity how function for if L the C in trivial satisfies scales the of b e l i e f clear i t could remains be to us are what interpreted. L s h o u l d be 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 n t h e s e n s e o f D e f i n i t i o n 3.2.6, and even of c h o o s i n g the w e i g h t s w ( i ) 1 for a l l p plays and t. undetermined, (and At consistent C(p,,—,p ) i t i s n e c e s s a r y t o have 'C(q ...,q ) = C(p,«q,,...,p 1 f n n L w(i) i=1 n «q n = ) 1 for n n all (p 1 f ...,p ),(qi,...,q ) e n n n ; but even that requirement is 130 not strong The of the enough t o c o m p l e t e l y following large partial variety determine result of p o o l i n g by t h e t e r m "non-local.'' Proposition 3.3.6 gives C. still another indication o p e r a t o r s w h i c h i s encompassed n Let 0 = {0,,...,0 } be f i n i t e , and assume that T:I1 —>n is a POB m pooling exists operator which i s prior-to-posterior coherent. a s e t {w(i,j,k)|1<i<n,1£j,k^m} of p o s i t i v e There constants such that T(p,,...,p for a l l p,,...,p n )(0 ) n k m n w(i,j,k) n n [p (0 ) ] j=1i=1 i j = e II and ke{ 1 ,... ,m}. Proof: Fix 0 e 0 and c o n s i d e r H(p,,...,p k n ) = T ( p , , . . . , p ) ( 0 ) as a n k n g r o u p homomorphism between n Then H(p ...,p ) 1 f n H = and with multiplication. n n H (p ) where H :I1—>(0,») i s d e f i n e d i=1 i i i i (p ) = H ( 1 , . . . , p ,...,1) i i i (p«q) = H (p)«H (q) whenever i i H (0,») for each i=1,...,n, p and q b e l o n g and t o II. we by have Each H can i m be decomposed further as H (p) i = n H ( p ( 0 )) j»1 i j D where H : i j 131 (0,»)—>(0,») i s defined by H (x) = H i j P-function whose v a l u e at 8 ( x ) , where xell i s that i i s x and 1 otherwise. j Now, H is a homomorphism on (0,»), and i t i s non-decreasing i j because T p r e s e r v e s it follows that H the (x) o r d e r i n g of b e l i e f s . = x w(i,j,k) Using Lemma 2.1.3, f o r some w ( i , j , k ) > 0, with ij k indicating facts, we To a p o s s i b l e d e p e n d e n c e on obtain the desired conclusion. conclude t h i s between the solution to the related s e c t i o n , we general so-called "bargaining general, given an r a n d o m i z e d d e c i s i o n r u l e s D*, u (_) i where u denotes the = i J7u Combining a l l these • would l i k e logarithmic opinion i n a manner w h i c h w i l l In 8 . k now be action t o draw a pool and problem." Nash's The space A and (a,0)S(da)b i are a space of i - t h player's utility (8)u(68), f u n c t i o n and b is a i axioms imply that the whichever i s appropriate. s o l u t i o n s , £*, n maximize two let or p o s t e r i o r d i s t r i b u t i o n , Nash's (1950) described. i prior parallel the symmetric product n [u ( $ ) - c ] i=1 i i are those Then which 1/n where c i 132 denotes the utility i - t h player's w h i c h he agree on will a choice s t a t u s quo have for i n the £. point, event that This maximization i . e . the the amount in group f a i l s i s subject to to u ($) i > c f o r a l l i=1,...,n (see Weerahandi & Zidek 1981 for further i details) . Now suppose 0 = {6i,...,6 } is finite, as in Proposition m 3.3.6. Assume f u r t h e r t h a t u (a,0) = 6(a,0) m c / £ b + i where 6(a,0) solution is the i j=1 Kronecker d e l t a f u n c t i o n . 1/n is the w(1) restricted to be would support ... = w(n) t o the Clearly, linear be n o n r a n d o m i z e d , the Nash 1/n. This operator relies by on could that choice the Equation pooling appealing an Observe optimal observation logarithmic (a)$(da)] i essentially from a s i m i l a r argument in turn measure. n 1/n II [b ( 0 ) ] , i =1 i general pooling (1975) w h i c h counting obtained = [Jb i=1 usual 0 which maximizes = j 1/n n i v denotes the which Then t h e n = (0Md0)] [/J6(a,0)$(da)b i=1 where i ) maximizes n n (0 if $ of a is quantity (3.3.1) lends with some a d d i t i o n a l recipe. be found to the unpublished for work of Stone's Bacharach contribution of 133 Madansky. optimal This work includes a t h e o r e m w h i c h shows t h a t t h e n Z w u ($). d e c i s i o n r u l e maximizes i=l reduction the which linear i s sketched opinion It Bacharach's r e s u l t implicitly utilities. This implicit Blackwell & Girshick conclusion. classical states there The i s only difficulty utility one In c e r t a i n may maximum j o i n t exp{0u not the different player and i n proving result however, players. that asserted deals with In that he the different situation, would have no a n d hence d e d u c i n g t h e problems, a propensity t h e 6 i n 0 f o r which t h e r e (MJP). of s t a t e s of nature, i . d e c i s i o n or e s t i m a t i o n be u s e d t o f i n d the i ' s represent presumably f o r the d i f f e r e n t propensity yield a r i s e s when a t h e o r e m o f c o m p a r i n g h i s own p r e f e r e n c e s functions function i s invoked Blackwell-Girshick nature, be n o t e d , would assumes t h e i n t e r c o m p a r a b i l i t y hypothesis (1954) paragraph should d e c i s i o n p r o b l e m where of i i i n the last pool. Indeed, the s o r t of I f , for example, i s the b {&) i = t h e 8 o f MJP i s t h e u n i q u e s o l u t i o n o f A ' ( 0 ) = - A(6)}, i _ u = n I a u . i =1 i i Weerahandi ix may This & Zidek well have has the c u r i o u s (1978) a that very P-functions, b . i the opinion i s that linear pool pointed i f t h e u 's a r e w i d e l y i low individual consequence propensity The c o r r e s p o n d i n g the joint out separated, as measured difficulty propensity in by t h e with function in 134 this case (say M) i s multi-modal. of low In p l a c e of a s i n g l e propensity relative representative u to each b , a f a m i l y of i M'S nonrepresentative (approximately the u 's themselves) is i obtained, these b each of high propensity relative t o e x a c t l y one of 's. i By D* extending in the the manner anomalies d e s c r i b e d circumvented. theory domain of advocated i n the In t h i s sketched the by W e e r a h a n d i previous extension, above, b pooling operator (0) & Zidek paragraph which b i and we the 0 o f MJP have j u s t amongst, to (1978), the easily be by described, this the the the ($) = J*b i i s r e p l a c e d by approximately, 0 can i s suggested i s r e p l a c e d by from (0)$(d0) i $ of MJP; i n the situation would l e a d t o a randomized widely M 'S. separated Nash The choice various i pooling operators unchanged under suggested this in this extension of chapter domain. d e r i v a t i o n w o u l d go t h r o u g h i n e x a c t l y t h e b ($)'s are P-functions too! i Finally, extend our whose range it should definition includes be added t h a t we of p r o p e n s i t y zero. would run into d i f f i c u l t y face in this situation In not since have not attempted Again, unlike their same way, function to case. remain fact, comprise Most d e r i v a t i o n s of t h i s in this is would that the the to those chapter problem encountered we in 135 conventional likelihood improvisation Bayesian functions analysis, have i s i n order. when disjoint the prior supports and and the some 136 IV. The present there i s no Along with doctors disagree However, t h e r e the to p r e d i c t experts. Granted, r e m a i n s t o be with sound that general results carry such to models In argues i s one -albeit techniques specified is "to in urgent of at the of point at the o u t s e t learn more" the seem t o be that from lies is cited the chronic w h i c h our difficult. case The ahead w h i c h we t h a t the purpose of or p e r h a p s t o make a have been but, says little doubt and validity of us. above, lack society 4 will development of This the Shapiro estimates... distributions." task experts individual models; reliability form idea e m p i r i c a l and the opinions in most mainly a witness distributions is s t i l l be 1971; concerning indirectly- that least Winkler when i n the a panel; than that belief. synthetic analytical Savage w h i c h of value would of c o m m u n i c a t i o n the a g g r e g a t i o n 1963; of always "there probability the paper reduced 241), judgments over of view "what t o do will of composite evidence p. average Sanders the (1977, of number with greater accuracy probed that and closure which Hogarth the been, convex in observed c o u l d say infinite the expressions i s great potential s t u d i e s ( c f . , e.g., a l . 1977,1979) one always the i f anything, to aggregate (1971), amongst FURTHER RESEARCH reinforced, way" has constitute various et best Savage choosing that work has "one quandary." of SUGGESTIONS FOR the of author dependable suffers makes difficulty is d e s c r i b e d where i t i s the decision forecast, but maker not to 137 take d e c i s i o n s . gross to finds that strategic be they are advantage. utility ineffective in s i t u a t i o n s competing solidly opinion departures communication factual members to a s s i s t one another e s t a b l i s h e d r u l e s as pools from will to show some the assumption help information the one another experts some of n e a r l y panel make linear or robustness N e v e r t h e l e s s , we a and more for to identical agree i s there to devise appropriate which relevant f o r example, a r e in decision-oriented tasks, f u n c t i o n s f o r the e x p e r t s . Savage t h a t t h e n e e d communication, where a g r o u p of with M o r e o v e r , even such logarithmic "reasonable" or or e x c e s s i v e deference, witnessed would expect the l a c k of c a n d o r exaggeration likely one A with methods of to h o n e s t l y share a l l it possible in thinking their for beliefs its through thoroughly. To a l a r g e degree, the problem scheme f o r weighing individual Efforts in this d i r e c t i o n Winkler (1971), inter solution to t h i s problem would the seem results vary with to recommendation seeking be that crucial to find they aspects opinions also have been made by alia. i s yet appropriate remains unsolved. Roberts However, no in sight. an the which parameter In w h i c h can currently interest. be made to The should sensitivity analysis the weight a l l o c a t i o n task. light to would of could, only decision of e x p e r t s of issue weights for a panel use the i n the a consensus and satisfactory fact, r e v e a l t h a t the of (1965) wholly have been made more c o m p l i c a t e d of S e c t i o n 3.3 6, of c h o o s i n g firm makers seem identify to the 138 In new a recent criteria based on for the the optimal under itself assessing idea of amongst is technical report, sample part interesting applied of t o see i n the we preserving introduced is that Nash's s o l u t i o n i s assumes t h a t and of yield the this group yields the process. will One i t i s found elicitation ideas two an priors It will when they be are context. list proved other experimental what t h e s e a w h i c h were r a i s e d i n t h e 1. We The where t h e the present Finally, g r o u p and from a s u p e r p o p u l a t i o n , a n a l o g u e of Wald's t h e o r y becomes the s o l u t i o n concepts only "subsampling." a (1982) group d e c i s i o n procedures. subsampling proposed Zidek in number of course Section semi-local pooling of unsettled the 2.3 issues discussion: that operator technical if T is any whose u n d e r l y i n g dogma function n "G" on linear [0,») opinion Proposition i s continuous, pool. 2.3.6, t h i s and, considering could conceivably 2. opinion local of In As the be noted pooling arbitrary is i n the condition way in weakened, Theorem 2.4.6, pool then T it on is also local second paragraph G seems rather which i t was used i f not was and artificial i n our removed a l t o g e t h e r . seen that the a v a i l a b l e E x t e r n a l l y Bayesian operator when (0,M) measure (Assumption non-negligible 2.4.5). proof, How? logarithmic only contains a following the small hence Can quasisets this 139 somewhat still, out irritating regularity eradicated? whether Externally I t would be p a r t i c u l a r l y there are Bayesian (allowing c o n d i t i o n be weakened o r , or are not procedures f o r negative weights w any than other the as long as i is finite 3. and u i s a c o u n t i n g - t y p e The problem of important better to find quasi-local logarithmic n L w =1) i= l i pool when 0 measure. determining which g e A 0 maximizes the product n P = n [/f i =1 arose a 1 -a w( i ) g du] i i n S e c t i o n 2.5 when we were t r y i n g density g optimized the expected a c (0,1). a, Renyi Information which order 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 t o s o l v e i t a n a l y t i c a l l y . could be v i e w e d functions. best as an a n a l o g u e However, developed i n Chapter seem t o be a h a r d 3 that of B a y e s ' arguments in formula favour w i t h i n t h e framework o f an the rule remain problem, (3.2.8) f o r updating P- o f i t s u s e c o u l d be axiomatic propensity functions. 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Title | Towards a consensus of opinion |
Creator |
Genest, Christian |
Publisher | University of British Columbia |
Date Issued | 1983 |
Description | This thesis addresses the problem of combining the prior density functions, f[sub=1],…,f[subn], of n individuals. In the first of two parts, various systems of axioms are developed which characterize successively the linear opinion pool, A(f[sub=1],...,f[sub=n]) = Σ[sup=n; sub= i=1] w[sub=i] f[sub=i] , and the logarithmic opinion pool, G(f[sub=1],…,f[sub=n]) = π[sup=n; sub= i=1] f[sup=α(i); sub=i] / ʃ π[sup=n; sub= i=1] dμ. It is first shown that A is the only pooling operator, T(f[sub=1],…,f[sub=n]), which is expressible as T(f[sub=1],…,f[sub=n]) (θ) = H(f[sub=1](θ),...,f[sub=n](θ), θ) for some function H which is continuous in its first n variables and satisfies H(0,...,0,θ) = 0 for μ- almost all θ. The regularity condition on H may be dispensed with if H does not depend on θ. This result leads to an impossibility theorem involving Madansky's axiom of External Bayesianity. Other consequences of this axiom of group rationality are also examined in some detail and yield a characterization of G as the only Externally Bayesian pooling operator of the form T(f[sub=1],…,f[sub=n])(θ) = H(f[sub=1],(θ),...,f[sub=n](θ))/ ʃH(f[sub=1],…,f[sub=n])dμ for some H:(0, ∞) —>(0, ∞). To prove this n result, it is necessary to introduce a "richness" condition on the underlying space of events, (θ,μ). Next, each opinion f[sub=i] is regarded as containing some "information" about θ and we look for a pooling operator whose expected information content is a maximum. The operator so obtained depends on the definition which is chosen; for example, Kullback-Leibler's definition entails the linear opinion pool, A. In the second part of the dissertation, it is argued that the domain of pooling operators should extend beyond densities. The notion of propensity function is introduced and examples are given which motivate this generalization; these include the well-known problem of combining P-values. A theorem of Aczel is adapted to derive a large class of pooling formulas which encompasses both A and G. A final characterization of G is given via the interpretation of betting odds, and the parallel between our approach and Nash's solution to the "bargaining problem" is discussed. |
Subject |
Consensus (Social sciences) -- Mathematical models |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-05-02 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
IsShownAt | 10.14288/1.0080161 |
URI | http://hdl.handle.net/2429/24289 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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- 831-1.0080161-fulltext.txt
- Citation
- 831-1.0080161.ris
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