Non-asymptotic Error Bounds for Sequential MCMC Methods

Abstract

Sequential MCMC methods are a class of stochastic numerical integration methods for target measures $\mu$ which cannot feasibly be attacked directly with standard MCMC methods due to the presence of multiple well-separated modes. The basic idea is to approximate the target distribution $\mu$ with a sequence of distributions $\mu_0,\ldots, \mu_n$ such that $\mu_n=\mu$ is the actual target distribution and such that $\mu_0$ is easy to sample from. The algorithm constructs a system of $N$ particles which sequentially approximates the measures $\mu_0$ to $\mu_n$. The algorithm is initialized with $N$ independent samples from $\mu_0$ and then alternates two types of steps, Importance Sampling Resampling and MCMC: In the Importance Sampling Resampling steps, a cloud of particles approximating $\mu_k$ is transformed into a cloud of particles approximating $\mu_{k+1}$ by randomly duplicating and eliminating particles in a suitable way depending on the relative density between $\mu_{k+1}$ and $\mu_{k}$. In the MCMC steps, particles move independently according to an MCMC dynamics for the current target distribution in order to adjust better to the changed environment.