Concentration rate and consistency of the posterior under monotonicity constraints

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.

References in corpus (5)

• Adaptive Bayesian multivariate density estimation with Dirichlet mixtures
• Confidence bands in density estimation
• Posterior convergence rates of Dirichlet mixtures at smooth densities
• Kullback Leibler property of kernel mixture priors in Bayesian density estimation
• The limit distribution of the $L_{\infty }$-error of Grenander-type estimators

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• Revisiting consistency of a recursive estimator of mixing distributions
• Rejoinder to discussions of "Frequentist coverage of adaptive nonparametric Bayesian credible sets"
• Discussion of "Frequentist coverage of adaptive nonparametric Bayesian credible sets"