Information-Theoretic Asymptotics of Bayes Methods

In the absence of knowledge of the true density function, Bayesian models take the joint density function for a sequence of n random variables to be an average of densities with respect to a prior. We examine the relative entropy distance Dn between the true density and the Bayesian density and show that the asymptotic distance is (d/2)(log n)+ c, where d is the dimension of the parameter vector. Therefore, the relative entropy rate Dn/n converges to zero at rate (log n)/n. The constant c, which we explicitly identify, depends only on the prior density function and the Fisher information matrix evaluated at the true parameter value. Consequences are given for density estimation, universal data compression, composite hypothesis testing, and stock-market portfolio selection.