Multivariate Density Estimation via Adaptive Partitioning (II): Posterior Concentration

TL;DR

This paper investigates Bayesian non-parametric density estimators based on adaptive binary partitions, demonstrating their ability to adapt to unknown density complexity and achieve optimal convergence rates regardless of dimensionality.

Contribution

It extends previous work by analyzing the posterior concentration rate of Bayesian piecewise constant density estimators, highlighting their adaptivity and dimension-independent convergence.

Findings

Posterior concentrates at an optimal rate regardless of dimension.

Bayesian estimators adapt to the true density's complexity.

Compared to sieve MLE, Bayesian methods achieve similar convergence without extra conditions.

Abstract

In this paper, we study a class of non-parametric density estimators under Bayesian settings. The estimators are piecewise constant functions on binary partitions. We analyze the concentration rate of the posterior distribution under a suitable prior, and demonstrate that the rate does not directly depend on the dimension of the problem. This paper can be viewed as an extension of a parallel work where the convergence rate of a related sieve MLE was established. Compared to the sieve MLE, the main advantage of the Bayesian method is that it can adapt to the unknown complexity of the true density function, thus achieving the optimal convergence rate without artificial conditions on the density.