Particle Markov Chain Monte Carlo Methods

Abstract

Summary. Markov chain Monte Carlo and sequential Monte Carlo methods have emerged as the two main tools to sample from high dimensional probability distributions. Although asymptotic convergence of Markov chain Monte Carlo algorithms is ensured under weak assumptions, the performance of these algorithms is unreliable when the proposal distributions that are used to explore the space are poorly chosen and/or if highly correlated variables are updated independently. We show here how it is possible to build efficient high dimensional proposal distributions by using sequential Monte Carlo methods. This allows us not only to improve over standard Markov chain Monte Carlo schemes but also to make Bayesian inference feasible for a large class of statistical models where this was not previously so. We demonstrate these algorithms on a non-linear state space model and a Lévy-driven stochastic volatility model.

Kernel methods in system identification, machine learning and function estimation: A survey

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The M etropolis– H astings Algorithm

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Pseudo-marginal Markov Chain Monte Carlo for Nonnegative Matrix Factorization

“…In most of the situations the marginal likelihood has no analytical form. Interestingly, when the marginal likelihood is not known, [2] and [1] have proposed to substitute the unknown marginal likelihood by an estimated one to compute the MH acceptance ratio, and it has been proved that under weak assumptions the algorithm leaves the target distribution p(M, θ M |X) invariant. The PMCMC algorithm described in Algorithm 1 is employed to simulate the posterior distribution p(M, θ M |X) for…”