Conditions for rapid mixing of parallel and simulated tempering on multimodal distributions

We give conditions under which a Markov chain constructed via parallel or simulated tempering is guaranteed to be rapidly mixing, which are applicable to a wide range of multimodal distributions arising in Bayesian statistical inference and statistical mechanics. We provide lower bounds on the spectral gaps of parallel and simulated tempering. These bounds imply a single set of sufficient conditions for rapid mixing of both techniques. A direct consequence of our results is rapid mixing of parallel and simulated tempering for several normal mixture models, and for the mean-field Ising model.Comment: Published in at http://dx.doi.org/10.1214/08-AAP555 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Forgetting the starting distribution in finite interacting tempering

Markov chain Monte Carlo (MCMC) methods are frequently used to approximately simulate high-dimensional, multimodal probability distributions. In adaptive MCMC methods, the transition kernel is changed "on the fly" in the hope to speed up convergence. We study interacting tempering, an adaptive MCMC algorithm based on interacting Markov chains, that can be seen as a simplified version of the equi-energy sampler. Using a coupling argument, we show that under easy to verify assumptions on the target distribution (on a finite space), the interacting tempering process rapidly forgets its starting distribution. The result applies, among others, to exponential random graph models, the Ising and Potts models (in mean field or on a bounded degree graph), as well as (Edwards-Anderson) Ising spin glasses. As a cautionary note, we also exhibit an example of a target distribution for which the interacting tempering process rapidly forgets its starting distribution, but takes an exponential number of steps (in the dimension of the state space) to converge to its limiting distribution. As a consequence, we argue that convergence diagnostics that are based on demonstrating that the process has forgotten its starting distribution might be of limited use for adaptive MCMC algorithms like interacting tempering

Swapping, tempering and equi-energy sampling on a selection of models in statistical mechanics

In dieser Arbeit werden drei Varianten des Metropolis-Hastings Algorithmus betrachtet. Simulated Tempering, Swapping und Equi-Energy Sampling sollen durch Hinzufügen eines Temperaturschritts die Konvergenzgeschwindigkeit dieser Algorithmen gegenüber dem zugrunde liegenden Metropolis-Hastings Algorithmus in Situationen verbessern, in denen dieser langsam gegen das gewünschte Wahrscheinlichkeitsmaß konvergiert. Es wird gezeigt, dass der Swapping Algorithmus im Generalized-Curie-Weiss Modell in polynomiell vielen Schritten konvergiert. Im Blume-Emery-Grifiths Modell ist die Konvergenz in einem Parameterbereich auch schnell, während sie in einem anderen Parameterbereich langsam ist. Auch für die Spingläser Random-Energy-Model und Generalized-Random-Energy-Model benötigen Simulated Tempering und Swapping exponentiell viele Schritte um nahe an die gewünschte Verteilung zu gelangen. Schließlich wird noch gezeigt, dass der Equi-Energy Algorithmus im Potts Modell langsam mischt

Small-world MCMC and convergence to multi-modal distributions: From slow mixing to fast mixing

We compare convergence rates of Metropolis--Hastings chains to multi-modal target distributions when the proposal distributions can be of ``local'' and ``small world'' type. In particular, we show that by adding occasional long-range jumps to a given local proposal distribution, one can turn a chain that is ``slowly mixing'' (in the complexity of the problem) into a chain that is ``rapidly mixing.'' To do this, we obtain spectral gap estimates via a new state decomposition theorem and apply an isoperimetric inequality for log-concave probability measures. We discuss potential applicability of our result to Metropolis-coupled Markov chain Monte Carlo schemes.Comment: Published at http://dx.doi.org/10.1214/105051606000000772 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org