Asymptotically Minimax Regret by Bayes Mixtures

We study the problems of data compression, gambling and prediction of a sequence $x^n=x_1x_2...x_n$ from an alphabet ${\cal X}$, in terms of regret and expected regret (redundancy) with respect to various smooth families of probability distributions. We evaluate the regret of Bayes mixture distributions compared to maximum likelihood, under the condition that the maximum likelihood estimate is in the interior of the parameter space. For general exponential families (including the non-i.i.d.\ case) the asymptotically mimimax value is achieved when variants of the prior of Jeffreys are used. %under the condition that the maximum likelihood estimate is in the interior of the parameter space. Interestingly, we also obtain a modification of Jeffreys prior which has measure outside the given family of densities, to achieve minimax regret with respect to non-exponential type families. This modification enlarges the family using local exponential tilting (a fiber bundle). Our conditions are confirmed for certain non-exponential families, including curved families and mixture families (where either the mixture components or their weights of combination are parameterized) as well as contamination models. Furthermore for mixture families we show how to deal with the full simplex of parameters. These results also provide characterization of Rissanen's stochastic complexity.