Semiparametric inference in mixture models with predictive recursion marginal likelihood

TL;DR

This paper introduces the predictive recursion marginal likelihood, a new Bayesian-inspired method for semiparametric mixture models that improves inference accuracy and computational efficiency over existing approaches.

Contribution

It develops a novel marginal likelihood approach based on predictive recursion, addressing uncertainty in structural parameters within mixture models.

Findings

The method shows convergence properties under model mis-specification.

Simulation results demonstrate improved density estimation performance.

Applications include mixed-effects models and empirical Bayes multiple testing.

Abstract

Predictive recursion is an accurate and computationally efficient algorithm for nonparametric estimation of mixing densities in mixture models. In semiparametric mixture models, however, the algorithm fails to account for any uncertainty in the additional unknown structural parameter. As an alternative to existing profile likelihood methods, we treat predictive recursion as a filter approximation to fitting a fully Bayes model, whereby an approximate marginal likelihood of the structural parameter emerges and can be used for inference. We call this the predictive recursion marginal likelihood. Convergence properties of predictive recursion under model mis-specification also lead to an attractive construction of this new procedure. We show pointwise convergence of a normalized version of this marginal likelihood function. Simulations compare the performance of this new marginal likelihood approach that of existing profile likelihood methods as well as Dirichlet process mixtures in density estimation. Mixed-effects models and an empirical Bayes multiple testing application in time series analysis are also considered.