A general minimax result for relative entropy

Suppose nature picks a probability measure P/sub /spl theta// on a complete separable metric space X at random from a measurable set P/sub /spl Theta//={P/spl theta/:/spl theta//spl isin//spl Theta/}. Then, without knowing /spl theta/, a statistician picks a measure Q on S. Finally, the statistician suffers a loss D(P/sub 0//spl par/Q), the relative entropy between P/sub /spl theta// and Q. We show that the minimax and maximin values of this game are always equal, and there is always a minimax strategy in the closure of the set of all Bayes strategies. This generalizes previous results of Gallager(1979), and Davisson and Leon-Garcia (1980).