Pseudo-extended Markov chain Monte Carlo

Sampling from posterior distributions using Markov chain Monte Carlo (MCMC) methods can require an exhaustive number of iterations, particularly when the posterior is multi-modal as the MCMC sampler can become trapped in a local mode for a large number of iterations. In this paper, we introduce the seudoextended MCMC method as a simple approach for improving the mixing of the MCMC sampler for multi-modal posterior distributions. The pseudo-extended method augments the state-space of the posterior using pseudo-samples as auxiliary variables. On the extended space, the modes of the posterior are connected, which allows the MCMC sampler to easily move between well-separated posterior modes. We demonstrate that the pseudo-extended approach delivers improved MCMC sampling over the Hamiltonian Monte Carlo algorithm on multi-modal posteriors, including Boltzmann machines and models with sparsity-inducing priors

Particle Metropolis-Hastings using gradient and Hessian information

Particle Metropolis-Hastings (PMH) allows for Bayesian parameter inference in nonlinear state space models by combining Markov chain Monte Carlo (MCMC) and particle filtering. The latter is used to estimate the intractable likelihood. In its original formulation, PMH makes use of a marginal MCMC proposal for the parameters, typically a Gaussian random walk. However, this can lead to a poor exploration of the parameter space and an inefficient use of the generated particles. We propose a number of alternative versions of PMH that incorporate gradient and Hessian information about the posterior into the proposal. This information is more or less obtained as a byproduct of the likelihood estimation. Indeed, we show how to estimate the required information using a fixed-lag particle smoother, with a computational cost growing linearly in the number of particles. We conclude that the proposed methods can: (i) decrease the length of the burn-in phase, (ii) increase the mixing of the Markov chain at the stationary phase, and (iii) make the proposal distribution scale invariant which simplifies tuning.Comment: 27 pages, 5 figures, 2 tables. The final publication is available at Springer via: http://dx.doi.org/10.1007/s11222-014-9510-

Metropolis Sampling

Monte Carlo (MC) sampling methods are widely applied in Bayesian inference, system simulation and optimization problems. The Markov Chain Monte Carlo (MCMC) algorithms are a well-known class of MC methods which generate a Markov chain with the desired invariant distribution. In this document, we focus on the Metropolis-Hastings (MH) sampler, which can be considered as the atom of the MCMC techniques, introducing the basic notions and different properties. We describe in details all the elements involved in the MH algorithm and the most relevant variants. Several improvements and recent extensions proposed in the literature are also briefly discussed, providing a quick but exhaustive overview of the current Metropolis-based sampling's world.Comment: Wiley StatsRef-Statistics Reference Online, 201

Getting Started with Particle Metropolis-Hastings for Inference in Nonlinear Dynamical Models

This tutorial provides a gentle introduction to the particle Metropolis-Hastings (PMH) algorithm for parameter inference in nonlinear state-space models together with a software implementation in the statistical programming language R. We employ a step-by-step approach to develop an implementation of the PMH algorithm (and the particle filter within) together with the reader. This final implementation is also available as the package pmhtutorial in the CRAN repository. Throughout the tutorial, we provide some intuition as to how the algorithm operates and discuss some solutions to problems that might occur in practice. To illustrate the use of PMH, we consider parameter inference in a linear Gaussian state-space model with synthetic data and a nonlinear stochastic volatility model with real-world data.Comment: 41 pages, 7 figures. In press for Journal of Statistical Software. Source code for R, Python and MATLAB available at: https://github.com/compops/pmh-tutoria