Information bounds for the risk of Bayesian predictions and the redundancy of universal codes

Several diverse problems have solutions in terms of an information-theoretic quantity for which we examine the asymptotics. Let Y₁, Y₂,..., Y_N be a sample of random variables with distribution depending on a (possibly infinite-dimensional) parameter θ. The maximum of the mutual information I_N = I(θ; Y₁, Y₂,..., Y_N) over choices of the prior distribution of θ provides a bound on the cumulative Bayes risk of prediction of the sequence of random variables for several choices of loss function. This same quantity is the minimax redundancy of universal data compression and the capacity of certain channels. General bounds for this mutual information are given. A special case concerns the estimation of binary-valued functions with Vapnik-Chervonenkis dimension d_vc, for which the information is bounded by d_vc log N. For smooth families of probability densities with a Euclidean parameter of dimension d, the information bound is (d/2) log N plus a constant. The prior density proportional to the square root of the Fisher information determinant is the unique continuous density that achieves a mutual information within o(1) of the capacity for large N. The Bayesian procedure with this prior is asymptotically minimax for the cumulative relative entropy risk.