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UBC Theses and Dissertations

Towards a consensus of opinion Genest, Christian 1983

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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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