Perfect Tempering

Abstract

Multimodal structures in the sampling density (e.g. two competing phases) can be a serious problem for traditional Markov Chain Monte Carlo (MCMC), because correct sampling of the different structures can only be guaranteed for infinite sampling time. Samples may not decouple from the initial configuration for a long time and autocorrelation times may be hard to determine. We analyze a suitable modification (C. J. Geyer and E. A. Thompson, J. Amer. Statist. Assoc., 90, 909, 1995) of the simulated tempering idea (E. Marinari and G. Parisi, Europhys. Lett. 19, 451, 1992), which has orders of magnitude smaller autocorrelation times for multimodal sampling densities and which samples all peaks of multimodal structures according to their weight. The method generates exact, i.e. uncorrelated, samples and thus gives access to reliable error estimates. Exact tempering is applicable to arbitrary (continuous or discreet) sampling densities and moreover presents a possibility to calculate integrals over the density (e.g. the partition function for the Boltzmann distribution), which are not accessible by usual MCMC.