Let X|μ∼Np(μ, vxI) and Y|μ∼Np(μ, vyI) be independent p-dimensional multivariate normal vectors with common unknown mean μ. Based on observing X=x, we consider the problem of estimating the true predictive density p(y|μ) of Y under expected Kullback–Leibler loss. Our focus here is the characterization of admissible procedures for this problem. We show that the class of all generalized Bayes rules is a complete class, and that the easily interpretable conditions of Brown and Hwang [Statistical Decision Theory and Related Topics (1982) III 205–230] are sufficient for a formal Bayes rule to be admissible.
"Admissible predictive density estimation." Ann. Statist. 36 (3) 1156 - 1170, June 2008. https://doi.org/10.1214/07-AOS506