How Regular Are Conjugate Exponential Families?
Müller-Funk U, Pukelsheim F
Given an exponential family of sampling distributions of orderk, one may construct in a natural way an exponential family of conjugate (that is, prior) distributions depending on ak-dimensional parametercand an additional weightw> 0. We compute the bias term by which the expectation of the sampling mean-value parameter under the conjugate distribution deviates from the conjugate parameterc. This bias term vanishes for regular exponential families, providing an appealing interpretation of the conjugate parametercas a ‘prior location' of the sampling mean-value parameter. By way of example we explore the extension of this approach to moments of higher order, in order to interprete the conjugate weightwas a ‘prior sample size'.
prior distributions, closedness under sampling, log-concavity, strong unimodality, mean-value parameter, Fisher information matrix, maximum likelihood estimate