  Simulated tempering (ST) is an established Markov chain Monte Carlo (MCMC)
method for sampling from a multimodal density $\pi(\theta)$. Typically, ST
involves introducing an auxiliary variable $k$ taking values in a finite subset
of $[0,1]$ and indexing a set of tempered distributions, say $\pi_k(\theta)
\propto \pi(\theta)^k$. In this case, 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), can disappoint. 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 the resulting estimator has a highly desirable property related
to the notion of effective sample size. We briefly report on the success of the
optimal combination in two modelling scenarios requiring reversible-jump MCMC,
where the na\"ive approach fails.
