Consistency results on nonparametric Bayesian estimation of level sets using spatial priors

Abstract

The main purpose of this paper is to propose a Bayesian nonparametric estimation of density’s level sets from n independent identically distributed observations. We prove the strong consistency of posterior distribution under a fairly general class of priors. This entails, in particular, that the Bayes estimate of the level set is consistent in terms of the Lebesgue measure of the symmetric difference. The considered priors put mass on piecewise-constant densities. The pieces on which the densities are constant are Voronoi tiles generated by a spatial point process. Using conditions similar to those considered in the density framework, we also obtain the consistency of the Bayesian level set estimates in the regression model. We present a simulation study using a special prior in the framework of density level set estimation. This allows us to visualise the numerical performances of the Bayesian estimate and to compare them to a plug-in level set estimate.