Bayes estimator for multinomial parameters and Bhattacharyya distances

We derive the Bayes estimator for the parameters of a multinomial distribution under two loss functions ($1-B$ and $1-B^2$) that are based on the Bhattacharyya coefficient $B(\vec{p},\vec{q}) = \sum{\sqrt{p_kq_k}}$. We formulate a non-commutative generalization relevant to quantum probability theory as an open problem. As an example application, we use our solution to find minimax estimators for a binomial parameter under Bhattacharyya loss ($1-B^2$).