Importance Tempering

Simulated tempering (ST) is an established Markov Chain Monte Carlo (MCMC) methodology for sampling from a multimodal density $\pi(\theta)$. The technique involves introducing an auxiliary variable k taking values in a finite subset of [0,1] and indexing a set of tempered distributions, say $\pi_k(\theta) \propto \pi(\theta)^k$. Small values of k encourage better mixing, but samples from $\pi$ are only obtained when the joint chain for $(\theta,k)$ reaches k=1. However, the entire chain can be used to estimate expectations under $\pi$ of functions of interest, provided that importance sampling (IS) weights are calculated. Unfortunately this method, which we call importance tempering (IT), has tended not work well in practice. This is partly because the most immediately obvious implementation is na\"ive and can lead to high variance estimators. We derive a new optimal method for combining multiple IS estimators and prove that this optimal combination has a highly desirable property related to the notion of effective sample size. The methodology is applied in two modelling scenarios requiring reversible-jump MCMC, where the na\"ive approach to IT fails spectacularly: model averaging in treed models, and model selection for mark--recapture data.