POSTERIOR CUMULANT RELATIONSHIPS IN BAYESIAN-INFERENCE INVOLVING THE EXPONENTIAL FAMILY

For BaYesian inference in one-parameter contexts where either the likelihood or the prior has an exponential family form, relationships are derived for Posterior moments and cumulants of (functions of) both the canonical and the expectation parameters. The identities exhibited generalize the simple relationships well known in the conjugate analysis case. Applications of these results are indicated in the areas of Bayesian robustness and approximation. In particular, results are obtained on the behavior of the posterior distribution for a large observation, generalizing work of Meeden and Isaacson.