The Bernstein-von Mises theorem for Semiparametric Mixtures

Abstract

Semiparametric mixture models are parametric models with latent variables. They are defined kernel, $p_{\hat{I}}¸(x|z)$, where z is the unknown latent variable, and $\hat{I}¸$ 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 $(\hat{I}¸,F)$. 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 $\hat{I}¸$ in the frailty model and the errors in variables model in the case were we put a generic prior on $\hat{I}¸$ and a species sampling process prior on $F$.