Abstract
Sequential Monte Carlo (SMC) methods are widely used for non-linear filtering purposes. Nevertheless the SMC scope encompasses wider applications such as estimating static model parameters so much that it is becoming a serious alternative to Markov-Chain Monte-Carlo (MCMC) methods. Not only SMC algorithms draw posterior distributions of static or dynamic parameters but additionally provide an estimate of the marginal likelihood. The tempered and time (TNT) algorithm, developed in the paper, combines (off-line) tempered SMC inference with on-line SMC inference for drawing realizations from many sequential posterior distributions without experiencing a particle degeneracy problem. Furthermore, it introduces a new MCMC rejuvenation step that is generic, automated and well-suited for multi-modal distributions. As this update relies on the wide heuristic optimization literature, numerous extensions are already available. The algorithm is notably appropriate for estimating Change-point models. As an example, we compare Change-point GARCH models through their marginal likelihoods over time.
Highlights
Sequential Monte Carlo (SMC) algorithm is a simulation-based procedure used in Bayesian framework for drawing distributions
The method encompasses the off-line AIS of Neal (1998), the on-line IBIS algorithm of Chopin (2002) and the RM method of Gilks and Berzuini (2001) that all arise as special cases in the SMC sampler theory (see Del Moral, Doucet, and Jasra (2006))
The tempered and time (TNT) algorithm benefits from the conjugacy of the tempered and the time domains to avoid particle degeneracies observed in the on-line methods
Summary
Sequential Monte Carlo (SMC) algorithm is a simulation-based procedure used in Bayesian framework for drawing distributions. It firstly iterates from the prior to the posterior distributions by means of a sequence of tempered posterior distributions It updates in the time dimension the slightly different posterior distributions by sequentially adding new observations, each SMC step providing all the forecast summary statistics relevant for comparing models. We start by emphasizing that the DiffeRential Evolution Adaptive Metropolis (DREAM, see Vrugt, ter Braak, Diks, Robinson, Hyman, and Higdon (2009)), the walk move (see Christen and Fox (2010)) and the stretch one (see Foreman-Mackey, Hogg, Lang, and Goodman (2013)) separately introduced in the statistic literature as generic Metropolis-Hastings proposals are standard mutation rules of the Differential Evolution (DE) optimization From this observation, we propose seven new MCMC updates based on the heuristic literature and emphasize that many other extensions are possible.
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