Moments for the canonical parameter of an exponential family under a conjugate distribution

Abstract

For exponential families, the moment generating function of a conjugate exponential family distribution for the canonical parameter can be conveniently written in terms of the corresponding normalising constant for the conjugate densities. This result can be used to find the moments of the canonical parameter and certain functions thereof. This is illustrated for the class of natural exponential families having a simple quadratic variance function. Two applications are outlined. The first is concerned with the analysis of dynamic generalised linear models, as put forward by West, Harrison & Migon (1985). The second refers to the calculations of the expected logarithmic divergence for exponential families under a conjugate distribution.