Admissible predictive density estimation

Let $X|musim N_p(mu,v_xI)$ and $Y|musim N_p(mu,v_yI)$ be independent $p$-dimensional multivariate normal vectors with common unknown mean $mu$. Based on observing $X=x$, we consider the problem of estimating the true predictive density $p(y|mu)$ 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.