Risk Bounds for Mixture Density Estimation on Compact Domains via the $h$-Lifted Kullback--Leibler Divergence

TL;DR

This paper introduces the $h$-lifted KL divergence for density estimation on compact domains, providing theoretical risk bounds and a practical estimation procedure with experimental validation.

Contribution

It extends existing risk bounds to non-strictly positive densities using the $h$-lifted KL divergence and develops a computational method for maximum $h$-lifted likelihood estimators.

Findings

Achieves an $ ext{O}(1/ oot{n}{})$ risk bound for density estimation.

Provides a practical algorithm for $h$-MLLE computation.

Experimental results support the theoretical risk bounds.

Abstract

We consider the problem of estimating probability density functions based on sample data, using a finite mixture of densities from some component class. To this end, we introduce the $h$-lifted Kullback--Leibler (KL) divergence as a generalization of the standard KL divergence and a criterion for conducting risk minimization. Under a compact support assumption, we prove an $\mathcal{O}(1/{\sqrt{n}})$ bound on the expected estimation error when using the $h$-lifted KL divergence, which extends the results of Rakhlin et al. (2005, ESAIM: Probability and Statistics, Vol. 9) and Li and Barron (1999, Advances in Neural Information ProcessingSystems, Vol. 12) to permit the risk bounding of density functions that are not strictly positive. We develop a procedure for the computation of the corresponding maximum $h$-lifted likelihood estimators ($h$-MLLEs) using the Majorization-Maximization framework and provide experimental results in support of our theoretical bounds.