Convergence of Sequential Markov Chain Monte Carlo Methods: I. Nonlinear Flow of Probability Measures

Sequential Monte Carlo Samplers are a class of stochastic algorithms for Monte Carlo integral estimation w.r.t. probability distributions, which combine elements of Markov chain Monte Carlo methods and importance sampling/resampling schemes. We develop a stability analysis by functional inequalities for a nonlinear flow of probability measures describing the limit behavior of the algorithms as the number of particles tends to infinity. Stability results are derived both under global and local assumptions on the generator of the underlying Metropolis dynamics. This allows us to prove that the combined methods sometimes have good asymptotic stability properties in multimodal setups where traditional MCMC methods mix extremely slowly. For example, this holds for the mean field Ising model at all temperatures. Spectral gap estimates, or, equivalently, Poincaré inequalities, as well as other related functional inequalities provide powerful tools for the study of convergence to equilibrium of reversible time-homogeneous Markov processes (see e.g. [9], [10], [11]). In particular, they have been successfully applied to analyze convergence properties of Markov Chain Monte Carlo (MCMC) methods based on reversible Markov chains (see e.g. [12]). The idea of MCMC methods is to produce approximate samples from a probability distribution μ by simulating for a sufficiently long time an ergodic Markov chain having μ as invariant measure. MCMC methods have become the standard to carry out Monte Carlo integrations with respect to complex probability distributions in many fields of applications, including in particular Bayesian statistics, statistical physics, and computational chemistry. We just refer the interested reader to [15] and [19] and references Date: December 3, 2006. 2000 Mathematics Subject Classification. 65C05, 60J25, 60B10, 47H20, 47D08.