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Towards a consensus of opinion

University of British Columbia

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

Text

2010-05-02

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http://hdl.handle.net/2429/24289

Mathematics

University of British Columbia

Graduate