Asymptotically minimax regret for models with hidden variables

Abstract

We study the problems of data compression, gambling and prediction of a string xn = x1x2...xn from an alphabet X, in terms of regret with respect to models with hidden variables including general mixture families. When the target class is a non-exponential family, a modification of Jeffreys prior which has measure outside the given family of densities was introduced to achieve the minimax regret [8], under certain regularity conditions. In this paper, we show that the models with hidden variables satisfy those regularity conditions, when the hidden variables' model is an exponential family. In paticular, we do not have to restrict the class of data strings so that the MLE is in the interior of the parameter space for the case of the general mixture family.