Importance sampling type estimators based on approximate marginal MCMC

TL;DR

This paper introduces importance sampling estimators based on approximate marginal MCMC for Bayesian models, offering advantages over delayed acceptance methods, with theoretical guarantees and promising experimental results.

Contribution

It proposes a novel importance sampling approach using approximate marginal MCMC, providing theoretical conditions, CLTs, and demonstrating practical benefits over existing methods.

Findings

The IS estimators are strongly consistent under minimal conditions.

The method achieves substantial efficiency gains over delayed acceptance schemes.

Experimental results show competitive performance even without parallelisation.

Abstract

We consider importance sampling (IS) type weighted estimators based on Markov chain Monte Carlo (MCMC) targeting an approximate marginal of the target distribution. In the context of Bayesian latent variable models, the MCMC typically operates on the hyperparameters, and the subsequent weighting may be based on IS or sequential Monte Carlo (SMC), but allows for multilevel techniques as well. The IS approach provides a natural alternative to delayed acceptance (DA) pseudo-marginal/particle MCMC, and has many advantages over DA, including a straightforward parallelisation and additional flexibility in MCMC implementation. We detail minimal conditions which ensure strong consistency of the suggested estimators, and provide central limit theorems with expressions for asymptotic variances. We demonstrate how our method can make use of SMC in the state space models context, using Laplace approximations and time-discretised diffusions. Our experimental results are promising and show that the IS type approach can provide substantial gains relative to an analogous DA scheme, and is often competitive even without parallelisation.