Posterior convergence rates of Dirichlet mixtures at smooth densities

TL;DR

This paper establishes the convergence rates of Bayesian posterior distributions using Dirichlet mixtures for smooth densities, matching the classical frequentist rate up to a logarithmic factor.

Contribution

It introduces a new general rate theorem for posterior convergence by combining covering arguments with prior probability bounds.

Findings

Posterior convergence rate is approximately n^{-2/5} with a logarithmic factor.

Derived bounds for Hellinger entropy numbers of the density class.

Established the optimality of the convergence rate for smooth densities.

Abstract

We study the rates of convergence of the posterior distribution for Bayesian density estimation with Dirichlet mixtures of normal distributions as the prior. The true density is assumed to be twice continuously differentiable. The bandwidth is given a sequence of priors which is obtained by scaling a single prior by an appropriate order. In order to handle this problem, we derive a new general rate theorem by considering a countable covering of the parameter space whose prior probabilities satisfy a summability condition together with certain individual bounds on the Hellinger metric entropy. We apply this new general theorem on posterior convergence rates by computing bounds for Hellinger (bracketing) entropy numbers for the involved class of densities, the error in the approximation of a smooth density by normal mixtures and the concentration rate of the prior. The best obtainable rate of convergence of the posterior turns out to be equivalent to the well-known frequentist rate for integrated mean squared error $n^{-2/5}$ up to a logarithmic factor.