A Bayesian Bootstrap for Mixture Models

TL;DR

This paper introduces a novel nonparametric Bayesian bootstrap method for mixture models, extending the traditional approach by replacing the point mass kernel with a continuous kernel, and provides theoretical guarantees and empirical illustrations.

Contribution

It develops a new Bayesian bootstrap for mixture models using a continuous kernel, expanding the applicability of the bootstrap in nonparametric Bayesian inference.

Findings

Proves convergence and exchangeability of the proposed algorithm

Demonstrates effectiveness through models and real data

Extends Bayesian bootstrap to mixture models with continuous kernels

Abstract

This paper proposes a new nonparametric Bayesian bootstrap for a mixture model, by developing the traditional Bayesian bootstrap. We first reinterpret the Bayesian bootstrap, which uses the P\'olya-urn scheme, as a gradient ascent algorithm which associated one-step solver. The key then is to use the same basic mechanism as the Bayesian bootstrap with the switch from a point mass kernel to a continuous kernel. Just as the Bayesian bootstrap works solely from the empirical distribution function, so the new Bayesian bootstrap for mixture models works off the nonparametric maximum likelihood estimator for the mixing distribution. From a theoretical perspective, we prove the convergence and exchangeability of the sample sequences from the algorithm and also illustrate our results with different models and settings and some real data.