SMC Samplers

Summary

This chapter covers SMC samplers , that is, particle algorithms that are able to track a sequence of probability distributions $\mathbb {P}_t(\mathrm {d}\theta )$, related by the recursion $\mathbb {P}_t(\mathrm {d}\theta ) = \ell _t G_t(\theta ) \mathbb {P}_{t-1}(\mathrm {d}\theta )$. Chapter 3 gave a few examples of applications of these algorithms; in some, one is genuinely interested in approximating each distribution in a given sequence, e.g., sequential Bayesian learning, in which case the sequence corresponds to the incorporation of more and more data; in others, one is interested in a single target distribution, but an artificial sequence is designed so as to be able to implement successfully an SMC sampler, and the sequence corresponds to increasing “temperatures”. Despite the different applications and appearances, the two approaches are not that different. We can think the former as “data tempering” and the latter as “temperature tempering” and in both situations sampling early distributions is easier than later ones.

When used to approximate a single distribution, SMC samplers represent an alternative to MCMC, with the following advantages: first, they offer a simple way to estimate the normalising constant of the target; this quantity is of interest in many problems, in particular in Bayesian model choice. Second, they are easier to parallelise. Third, it is reasonably easy to develop adaptive SMC samplers; that is algorithms which calibrate their tuning parameters automatically, using the current sample of particles.

We start by describing a Feynman-Kac model that formalises a large class of SMC samplers, based on invariant kernels. We then provide practical recipes to obtain good performance in the following scenarios: Bayesian sequential learning, tempering, rare-event simulation, and likelihood-free inference. Finally, we explain how to further extend SMC samplers by using more general Markov kernels.