On Rates of Convergence for Posterior Distributions in Infinite-dimensional Models

This paper introduces a new approach to the study of rates of convergence for posterior distributions. It is a natural extension of a recent approach to the study of Bayesian consistency. In particular , we improve on current rates of convergence for models including the mixture of Dirichlet process model and the random Bernstein polynomial model. 1. Introduction. Recently, there have been many contributions to the theory of Bayesian consistency for infinite-dimensional models. Most of these adopt the " frequentist " (or " what if ") approach, which consists of generating independent data from a " true " fixed density f 0 and checking whether the sequence of posterior distributions accumulates in Hellinger neighborhoods of f 0. The determination of sufficient conditions for Hellinger consistency has been the main goal of a number of recent papers such as, for example, [1, 2, 5] and [12]. A summary is provided in [8]. Their results rely upon the use of uniformly consistent tests, combined with the construction of suitable sieves and computation of metric entropies. An alternative method for solving the problem can be found in [14], where a sufficient condition in terms of the summability of prior probabilities is provided. Here, we consider the allied problem of determining rates of convergence, that is, the determination of a sequence (ε n) n≥1 such that ε n ↓ 0 and