Lr. Pericchi et al., POSTERIOR CUMULANT RELATIONSHIPS IN BAYESIAN-INFERENCE INVOLVING THE EXPONENTIAL FAMILY, Journal of the American Statistical Association, 88(424), 1993, pp. 1419-1426
For BaYesian inference in one-parameter contexts where either the like
lihood 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 ge
neralize the simple relationships well known in the conjugate analysis
case. Applications of these results are indicated in the areas of Bay
esian robustness and approximation. In particular, results are obtaine
d on the behavior of the posterior distribution for a large observatio
n, generalizing work of Meeden and Isaacson.