Exponential and Bayesian conjugate families: Review and extensions

The notion of a conjugate family of distributions plays a very important role in the Bayesian approach to parametric inference. One of the main features of such a family is that it is closed under sampling, but a conjugate family often provides prior distributions which are tractable in various other respects. This paper is concerned with the properties of conjugate families for exponential family models. Special attention is given to the class of natural exponential families having a quadratic variance function, for which the theory is particularly fruitful. Several classes of conjugate families have been considered in the literature and here we describe some of their most interesting features. Relationships between such classes are also discussed. Our aim is to provide a unified approach to the theory of conjugate families for exponential family likelihoods. An important aspect of the theory concerns reparameterisations of the exponential family under consideration. We briefly review the concept of a conjugate parameterisation, which provides further insight into many of the properties discussed throughout the paper. Finally, further implications of these results for Bayesian conjugate analysis of exponential families are investigated.