Kullback Leibler property of kernel mixture priors in Bayesian density estimation

TL;DR

This paper establishes verifiable conditions under which kernel mixture priors in Bayesian density estimation possess the Kullback-Leibler property, ensuring posterior consistency across various practical kernels.

Contribution

It provides a broad set of sufficient conditions for the Kullback-Leibler property of kernel mixture priors, applicable to many common kernel types in Bayesian density estimation.

Findings

Kullback-Leibler property holds for normal, t, histogram, gamma, Weibull kernels under certain conditions.

Conditions are easily verifiable at the true density, aiding practical application.

Results extend previous work to a wider class of kernels used in practice.

Abstract

Positivity of the prior probability of Kullback-Leibler neighborhood around the true density, commonly known as the Kullback-Leibler property, plays a fundamental role in posterior consistency. A popular prior for Bayesian estimation is given by a Dirichlet mixture, where the kernels are chosen depending on the sample space and the class of densities to be estimated. The Kullback-Leibler property of the Dirichlet mixture prior has been shown for some special kernels like the normal density or Bernstein polynomial, under appropriate conditions. In this paper, we obtain easily verifiable sufficient conditions, under which a prior obtained by mixing a general kernel possesses the Kullback-Leibler property. We study a wide variety of kernel used in practice, including the normal, $t$, histogram, gamma, Weibull densities and so on, and show that the Kullback-Leibler property holds if some easily verifiable conditions are satisfied at the true density. This gives a catalog of conditions required for the Kullback-Leibler property, which can be readily used in applications.