Importance tempering

Robert Gramacy, Richard Samworth, Ruth King

Research output: Contribution to journalArticlepeer-review

25 Citations (Scopus)


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)ae 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 < ve 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 < ve approach fails.

Original languageEnglish
Pages (from-to)1-7
Number of pages7
JournalStatistics and Computing
Issue number1
Publication statusPublished - Jan 2010


  • Simulated tempering
  • Importance sampling
  • Markov chain Monte Carlo (MCMC)
  • Metropolis-coupled MCMC


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