A priori truncation method for posterior sampling from homogeneous normalized completely random measure mixture models

TL;DR

This paper introduces a truncation method for normalized homogeneous completely random measure mixture models, enabling efficient posterior sampling and demonstrating versatility in density estimation and covariate incorporation.

Contribution

A novel a priori truncation approach for posterior sampling in normalized completely random measure models, with theoretical convergence guarantees and practical applications.

Findings

Proved convergence of the truncation approximation.

Demonstrated good performance in density estimation.

Extended the model to include covariates for regression and clustering.

Abstract

This paper adopts a Bayesian nonparametric mixture model where the mixing distribution belongs to the wide class of normalized homogeneous completely random measures. We propose a truncation method for the mixing distribution by discarding the weights of the unnormalized measure smaller than a threshold. We prove convergence in law of our approximation, provide some theoretical properties and characterize its posterior distribution so that a blocked Gibbs sampler is devised. The versatility of the approximation is illustrated by two different applications. In the first the normalized Bessel random measure, encompassing the Dirichlet process, is introduced; goodness of fit indexes show its good performances as mixing measure for density estimation. The second describes how to incorporate covariates in the support of the normalized measure, leading to a linear dependent model for regression and clustering.