2013-11-21: Nial Friel & Jimmy Olsson

Post date: 08-Nov-2013 09:29:35

Partial ordering of inhomogeneous Markov chains with applications to Markov Chain Monte Carlo methods

(joint work with Florian Maire and Randal Douc)

In this talk we will discuss the asymptotic variance of sample path averages for inhomogeneous Markov chains that evolve alternatingly according to two different π-reversible Markov transition kernels. More specifically, we define a partial ordering over the pairs of π-reversible Markov kernels, which allows us to compare directly the asymptotic variances for the inhomogeneous Markov chains associated with each pair. As an important application we use this result for comparing different data-augmentation-type Metropolis Hastings algorithms. In particular, we compare some pseudo-marginal algorithms and propose a novel exact algorithm, referred to as the random refreshment algorithm, which is more efficient, in terms of asymptotic variance, than the Grouped Independence Metropolis Hastings algorithm and has a computational complexity that does not exceed that of the Monte Carlo Within Metropolis algorithm. Finally, we provide a theoretical justification of the Carlin and Chib algorithm used in model selection.

* 16h10: Nial Friel

Convergence of Markov chains with approximate transition kernels with application to intractable likelihood statistical models.

(joint work with Pierre Alquier (UCD), Aidan Boland (UCD) and Richard Everitt (University of Reading))

Monte Carlo algorithms often aim to draw from a distribution \pi by simulating a Markov chain with transition kernel P such that \pi is invariant under P. However, there are many situations for which it is impractical or impossible to draw from the transition kernel P. For instance, this is the case with massive datasets, where is it prohibitively expensive to calculate the likelihood and is also the case for intractable likelihood models such as those found in spatial statistics and network analysis. A natural approach in these cases is to replace P by an approximation \hat{P}. Using theory from the stability of Markov chains we explore a variety of situations where it is possible to quantify how 'close' the invariant distribution \hat{\pi} with transition kernel \hat{P} is to \pi. We apply these results to several examples from spatial statistics and network analysis.