(3) particle filters and population Monte Carlo.
Nowadays more-or-less everyone in astronomy has heard of Markov Chain Monte Carlo (MCMC), and a substantial fraction of astronomers will have either run an MCMC chain themselves or at least coauthored a paper for which one was run. However, few astronomers will be aware of the other popular contemporary tool for exploring the Bayesian posterior: the particle filter (cf. population Monte Carlo [PMC], cf. sequential Monte Carlo [SMC]: Gordon et al. 1993, Doucet et al. 2000, Del Moral et al. 2006, Andrieu et al. 2010). Whereas the MCMC method aims to approximate the posterior upon convergence towards the stationary distribution of an ergodic Markov Chain, typically represented by a single ‘particle’ begun from an arbitrary starting state, the particle filter method begins with a diffuse population of many particles which it aims to evolve towards the true posterior sequentially via the key operations of reweighting and importance sampling.
By far the greatest strength of particle filtering methods emerges in application to the analysis of time series in which the system is often modelled as possessing a hidden time series of latent state variables drawn as the realisation a stochastic evolutionary rule, but observable only subject to some imprecise measurement process (typically iid). The classic scenario is that of tracking a moving object within a noisy field (cf. Yilmaz et al. 2006 for a review); and, of course, for modelling financial time series (cf. Lopes & Tsay 2010 for a review)! It is within this class of time series model—whereby the dimensionality and degeneracy of the parameter space increase at pace with data dimension—that particle filter methods offer their most ‘obvious’ advantage over MCMC methods: that we can explore the posterior within having to search (perhaps fruitlessly) for an effective proposal.
With astronomical time series analyses moving gradually towards the complex stochastic process models prominent in quantitative finance (see e.g. Bailer-Jones et al. 2012, Meyer et al. 2014, Sobolewska et al. 2014) it seems inevitable that particle filtering techniques will become a necessary tool of future astrostatistics. For an example of the power and sophistication of current particle filtering ideas the interested reader might look towards Lindsten et al.’s (2014) “Particle Gibbs with Ancestor Sampling” and its application to tracking an epidemiological model with non-Markovian latent time series.
Even outside the time series domain the particle filter methodology has proven useful for marginal likelihood computation in Bayesian model selection—this was the goal of its most substantial past use in cosmology: Wraith et al. 2009, Kilbinger et al. 2010—as well as posterior exploration in the big data framework as we’ll discuss further in part 5 (Chopin et al. 2002) and for approximate Bayesian computation as we’ll discuss in part 4 (Del Moral et al. 2011).
Visualisation of a particle filter (credit: Michael Pfeiffer).
References/Notes. An excellent introduction to particle filtering methods with applications in the physical sciences is available in the “Monte Carlo Strategies in Scientific Computing” by J.S. Liu.