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 Towards a consensus of opinion
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Towards a consensus of opinion Genest, Christian
Abstract
This thesis addresses the problem of combining the prior density functions, f[sub=1],…,f[subn], of n individuals. In the first of two parts, various systems of axioms are developed which characterize successively the linear opinion pool, A(f[sub=1],...,f[sub=n]) = Σ[sup=n; sub= i=1] w[sub=i] f[sub=i] , and the logarithmic opinion pool, G(f[sub=1],…,f[sub=n]) = π[sup=n; sub= i=1] f[sup=α(i); sub=i] / ʃ π[sup=n; sub= i=1] dμ. It is first shown that A is the only pooling operator, T(f[sub=1],…,f[sub=n]), which is expressible as T(f[sub=1],…,f[sub=n]) (θ) = H(f[sub=1](θ),...,f[sub=n](θ), θ) for some function H which is continuous in its first n variables and satisfies H(0,...,0,θ) = 0 for μ almost all θ. The regularity condition on H may be dispensed with if H does not depend on θ. This result leads to an impossibility theorem involving Madansky's axiom of External Bayesianity. Other consequences of this axiom of group rationality are also examined in some detail and yield a characterization of G as the only Externally Bayesian pooling operator of the form T(f[sub=1],…,f[sub=n])(θ) = H(f[sub=1],(θ),...,f[sub=n](θ))/ ʃH(f[sub=1],…,f[sub=n])dμ for some H:(0, ∞) —>(0, ∞). To prove this n result, it is necessary to introduce a "richness" condition on the underlying space of events, (θ,μ). Next, each opinion f[sub=i] is regarded as containing some "information" about θ and we look for a pooling operator whose expected information content is a maximum. The operator so obtained depends on the definition which is chosen; for example, KullbackLeibler's definition entails the linear opinion pool, A. In the second part of the dissertation, it is argued that the domain of pooling operators should extend beyond densities. The notion of propensity function is introduced and examples are given which motivate this generalization; these include the wellknown problem of combining Pvalues. A theorem of Aczel is adapted to derive a large class of pooling formulas which encompasses both A and G. A final characterization of G is given via the interpretation of betting odds, and the parallel between our approach and Nash's solution to the "bargaining problem" is discussed.
Item Metadata
Title 
Towards a consensus of opinion

Creator  
Publisher 
University of British Columbia

Date Issued 
1983

Description 
This thesis addresses the problem of combining the prior
density functions, f[sub=1],…,f[subn], of n individuals. In the first of
two parts, various systems of axioms are developed which
characterize successively the linear opinion pool, A(f[sub=1],...,f[sub=n])
= Σ[sup=n; sub= i=1] w[sub=i] f[sub=i] , and the logarithmic opinion pool, G(f[sub=1],…,f[sub=n]) = π[sup=n; sub= i=1] f[sup=α(i); sub=i] / ʃ π[sup=n; sub= i=1] dμ. It is first shown that A is the only pooling operator, T(f[sub=1],…,f[sub=n]), which is expressible as
T(f[sub=1],…,f[sub=n]) (θ) = H(f[sub=1](θ),...,f[sub=n](θ), θ) for some function H which is continuous in its first n variables and satisfies
H(0,...,0,θ) = 0 for μ almost all θ. The regularity condition
on H may be dispensed with if H does not depend on θ. This
result leads to an impossibility theorem involving Madansky's
axiom of External Bayesianity. Other consequences of this axiom
of group rationality are also examined in some detail and yield
a characterization of G as the only Externally Bayesian pooling
operator of the form T(f[sub=1],…,f[sub=n])(θ) = H(f[sub=1],(θ),...,f[sub=n](θ))/
ʃH(f[sub=1],…,f[sub=n])dμ for some H:(0, ∞) —>(0, ∞). To prove this n
result, it is necessary to introduce a "richness" condition on
the underlying space of events, (θ,μ). Next, each opinion f[sub=i] is
regarded as containing some "information" about θ and we look for a pooling operator whose expected information content is a maximum. The operator so obtained depends on the definition which is chosen; for example, KullbackLeibler's definition entails the linear opinion pool, A.
In the second part of the dissertation, it is argued that the domain of pooling operators should extend beyond densities. The notion of propensity function is introduced and examples are given which motivate this generalization; these include the wellknown problem of combining Pvalues. A theorem of Aczel is adapted to derive a large class of pooling formulas which encompasses both A and G. A final characterization of G is given via the interpretation of betting odds, and the parallel between our approach and Nash's solution to the "bargaining problem" is discussed.

Genre  
Type  
Language 
eng

Date Available 
20100502

Provider 
Vancouver : University of British Columbia Library

Rights 
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.

DOI 
10.14288/1.0080161

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Campus  
Scholarly Level 
Graduate

Aggregated Source Repository 
DSpace

Item Media
Item Citations and Data
Rights
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.