Convergence Rates of Posterior Distributions

If the distribution P is considered random and distributed according to , as it is in Bayesian inference, then the posterior distribution is the conditional distribution of P given the observations. The prior is, of course, a measure on some σ-field on and we must assume that the expressions in the display are well defined. In particular, we assume that the map x p → p x is measurable for the product σ-field on × . It will be silently understood that the sets of which we compute prior or posterior measures are measurable. In this paper we study the frequentist properties of the posterior distribution as n → ∞, assuming that the observations are a random sample from some fixed measure P0. In particular, we study the rate at which this random distribution converges to P0. The posterior is said to be consistent if, as a random measure, it concentrates on arbitrarily small neighborhoods of P0, with probability tending to 1 or almost surely, as n → ∞. We study the rate at which such neighborhoods may decrease to zero meanwhile still capturing most of the posterior mass. If = Pθ θ ∈ is parametrized by a parameter θ, then usually the prior is constructed by putting a measure on the parameter set . If is a