Asymptotically minimax regret for exponential families

We study the problem of data compression, gambling and prediction of a sequence x = x1x2...xn from a certain alphabet X , in terms of regret and redundancy with respect to a general exponential family. In particular, we evaluate the regret of the Bayes mixture density and show that it asymptotically achieves their minimax values when variants of Jeffreys prior are used. Keywords— universal coding, Bayes mixture, Jeffreys prior, exponential family