# Importance Tempering

### Robert B. Gramacy, Richard, J. Samworth and Ruth King

### Universities of Cambridge and St. Andrews

## 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.