Events
CRiSM Seminar
Dr Robert Gramacy, Statistical Laboratory Cambridge
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) = 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 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
for mark--recapture data.