The Bernstein-von Mises theorem for Semiparametric Mixtures

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 $F$.