# Importance Tempering

## Summary

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 mising, 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 to work well in practice. This is partly because the most immediate obvious implementation is naive 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 naive approach to IT fails: model averaging in treed models, and model selection in mark-recapture data.

### Keywords:

Simulated tempering, importance sampling, Markov chain Monte Carlo (MCMC), Metropolis-coupled MCMC, reversible jump MCMC, treed model, Gaussian process, mark-recapture data, model selection, model averaging.