Convergence rates for Bayesian density estimation of infinite-dimensional exponential families

TL;DR

This paper investigates the convergence rates of Bayesian methods for density estimation in infinite-dimensional exponential families, proposing priors that adapt to unknown smoothness levels and achieve optimal minimax rates.

Contribution

It introduces novel priors and truncation techniques that enable adaptive Bayesian density estimation with optimal convergence rates in Sobolev classes.

Findings

Gaussian priors achieve optimal rates when p is known.

Mixture of normals priors adapt to unknown smoothness levels.

Series truncation methods improve computational feasibility.

Abstract

We study the rate of convergence of posterior distributions in density estimation problems for log-densities in periodic Sobolev classes characterized by a smoothness parameter p. The posterior expected density provides a nonparametric estimation procedure attaining the optimal minimax rate of convergence under Hellinger loss if the posterior distribution achieves the optimal rate over certain uniformity classes. A prior on the density class of interest is induced by a prior on the coefficients of the trigonometric series expansion of the log-density. We show that when p is known, the posterior distribution of a Gaussian prior achieves the optimal rate provided the prior variances die off sufficiently rapidly. For a mixture of normal distributions, the mixing weights on the dimension of the exponential family are assumed to be bounded below by an exponentially decreasing sequence. To avoid the use of infinite bases, we develop priors that cut off the series at a sample-size-dependent truncation point. When the degree of smoothness is unknown, a finite mixture of normal priors indexed by the smoothness parameter, which is also assigned a prior, produces the best rate. A rate-adaptive estimator is derived.