Concentration rate and consistency of the posterior under monotonicity constraints

TL;DR

This paper investigates Bayesian estimation of monotone decreasing densities, establishing the posterior's concentration rate near the minimax rate and proving its consistency for point-wise and sup norm losses.

Contribution

It derives the concentration rate of the posterior under monotonicity constraints for Dirichlet process and finite mixture priors, showing near-minimax optimality.

Findings

Posterior concentrates at rate (n/log(n))^{-1/3}

Posterior is consistent for point-wise loss

Posterior is consistent for sup norm

Abstract

In this paper, we consider the well known problem of estimating a density function under qualitative assumptions. More precisely, we estimate monotone non increasing densities in a Bayesian setting and derive concentration rate for the posterior distribution for a Dirichlet process and finite mixture prior. We prove that the posterior distribution based on both priors concentrates at the rate $(n/\log(n))^{-1/3}$, which is the minimax rate of estimation up to a \log(n)$ factor. We also study the behaviour of the posterior for the point-wise loss at any fixed point of the support the density and for the sup norm. We prove that the posterior is consistent for both losses.