The Bernstein-von Mises theorem for Semiparametric Mixtures

TL;DR

This paper establishes a Bernstein-von Mises theorem for semiparametric mixture models, demonstrating Bayesian consistency and efficiency in estimating parameters within complex latent variable frameworks.

Contribution

It proves a general Bernstein-von Mises theorem for semiparametric mixtures and provides practical tools for verifying efficiency in specific models.

Findings

Proved Bayesian consistency for semiparametric mixture models.

Established a general Bernstein-von Mises theorem in this context.

Applied results to frailty and errors-in-variables models.

Abstract

Semiparametric mixture models are parametric models with latent variables. They are defined kernel, $p_\theta(x | z)$, where z is the unknown latent variable, and $\theta$ is the parameter of interest. We assume that the latent variables are an i.i.d. sample from some mixing distribution $F$. A Bayesian would put a prior on the pair $(\theta, F)$. We prove consistency for these models in fair generality and then study efficiency. We first prove an abstract Semiparametric Bernstein-von Mises theorem, and then provide tools to verify the assumptions. We use these tools to study the efficiency for estimating $\theta$ in the frailty model and the errors in variables model in the case were we put a generic prior on $\theta$ and a species sampling process prior on $F$.