  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.
