  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
($1-B^2$).
