An O(log<sub>2</sub> N) SMC<sup>2</sup> Algorithm on Distributed Memory with an Approx. Optimal L-Kernel
Conor Rosato, Alessandro Varsi, Joshua Murphy, Simon Maskell
Abstract
Calibrating statistical models using Bayesian inference often requires both accurate and timely estimates of parameters of interest. Particle Markov Chain Monte Carlo (p-MCMC) and Sequential Monte Carlo Squared (SMC <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) are two methods that use an unbiased estimate of the log-likelihood obtained from a particle filter (PF) to evaluate the target distribution. P-MCMC constructs a single Markov chain which is sequential by nature so cannot be readily parallelized using Distributed Memory (DM) architectures. This is in contrast to SMC <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> which includes processes, such as importance sampling, that are described as embarrassingly parallel. However, difficulties arise when attempting to parallelize resampling. None-the-less, the choice of backward kernel, recycling scheme and compatibility with DM architectures makes SMC <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> an attractive option when compared with p-MCMC. In this paper, we present an SMC <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> framework that includes the following features: an optimal (in terms of time complexity) $\mathcal{O}(\log_2 N)$ parallelization for DM architectures, an approximately optimal (in terms of accuracy) backward kernel, and an efficient recycling scheme. On a cluster of 128 DM processors, the results on a biomedical application show that SMC <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> achieves up to a 70× speed-up vs its sequential implementation. It is also more accurate and roughly 54× faster than p-MCMC. A GitHub link is given which provides access to the code.