Duality Induced from Conjugacy in the Curved Exponential Family

Abstract

A class of curved exponential families whose likelihood function admits the conjugate analysis is derived, and its duality is explored. We show that conjugacy yields the existence of sufficient statistics as well as duality. Extended versions of the mean and the canonical parameters can be defined, which shed a new light on duality and the conjugate analysis in the exponential family. As a result, an essential reason is revealed as to why a common prior density can be conjugate for different sampling densities, as in the case of a gamma prior density which is conjugate for the Poisson and the gamma sampling densities. The least information property of the conjugate analysis is explained, which is compatible with the minimax property of the generalized linear model. We also derive dual Pythagorean relationships with respect to posterior risks to show the optimality of the Bayes estimator.