Conjugate Priors for Exponential Families Having Quadratic Variance Functions

Conjugate Parameterizations for Natural Exponential Families

“…We also define the key idea of a conjugate parameterization and establish some basic results. In Section 3 we discuss transformations of the canonical parameter of a natural exponential family and extend the result of Consonni and Veronese (1992) mentioned earlier. We also explore conditions under which, for a given arbitrary parameterization, the standard conjugate family contains the corresponding Jeffreys' prior.…”

“…Under what conditions does this family coincide with the conjugate family induced on X by the standard conjugate family of 8? For regular natural exponential families, this is equivalent to asking when the standard conjugate family of X preserves the linearity property of the posterior expectation of the mean parameter p. Consonni and Veronese (1992) have recently provided an answer to this question for the particular case where X is the mean parameter itself. They have shown that for oneparameter natural exponential families, the standard conjugate family of p and the conjugate family induced on p by 8 coincide if and only if the natural exponential family has a quadratic variance function (i.e., a polynomial of degree at most 2 in p ) .…”

“…We note that Cx(F) is usually referred to as the natural conjugate family to the family 3. But because this might cause some confusion in the present context, we follow Consonni and Veronese (1992) and refer to Cx(F) instead as the standard conjugate family. An important property of standard conjugate families is that they are closed under sampling, in the sense that formal updating of a prior distribution in Cx(F) via Bayes's theorem yields a posterior distribution that also belongs to Cx(F).…”

“…Consonni andVeronese (1992, p. 1124) provided a partial answer by dealing with the case p -8 for one-parameter exponential families.…”

“…In the context of Bayesian parametric inference, a family of distributions whose densities have the structure of the likelihood kernel is called standard conjugate (Consonni and Veronese 1992). One of the main features of such a family is that it is closed under sampling, but this condition does not characterize the standard conjugate family, as shown by Diaconis and Ylvisaker (1979).…”