Priors for Exponential Families Which Maximize the Association between Past and Future Observations

Abstract

Throughout the present paper, {X_n} denotes a sequence of random quantities which are regarded as exchangeable, and which are assessed with a probability measure P(•) which is a member of the mixture-exponential family. To be precise, it will be presumed that the assessment P(•) for any finite subsequence (X₁,…,X_n) can be represented using the product of an identical non-degenerate parametric measure for each X_i, P_θ (•)= P(•θ=θ), determined by

$$ {\rm{d}}{{\rm{P}}_{\rm{\theta }}} = \exp \{ {\rm{\theta x}} - {\rm{M}}({\rm{\theta }})\} {\rm{d\mu }} $$

((1.1))

μ being a σ-finite measure on the class B of Borel sets of IR. It will always be assumed that the interior X° of the convex hull X of the support of μ (in symbols:suρp(μ))is a nonempty open set (interval) in IR and that {P_θ;θεΘ} is a regular exponential family (cf. Barndorff - Nielsen 1978, p.116). The latter condition implies that Θ = {θ:M(θ)<∞ } is an open interval in IR. Moreover, we will suppose that the set of the logically possible values of θ coincides with θ. Given such a particular frame, the present paper deals with the choice of a prior for (1.1); an excellent treatment of the same topic is included in Diaconis and Ylvisaker (1979, 1985). Our approach bases itself on the obvious remark that the choice of a prior establishes the strength of the dependence among the elements of the sequence {X_n} and, consequently, the strength of the influence exercised by experience on our future predictions. This subjective standpoint is skilfully expounded in de Finetti (1937).