Workshop
"Markov chain Monte Carlo (MCMC) methods"
26th of november 2014

Abstract
Nous présenterons dans ce travail un panorama des méthodes de méthodes MCMC adaptatives. Nous couvrirons en particulier les méthodes élémentaires (Adaptive Random Walk Metropolis) avant de nous intéresser à des méthodes beaucoup plus sophistiquées (méthodes MCMC en interactions). Nous illustrerons ces méthodes dans différentes applications.

Abstract
A Kernel Adaptive Metropolis-Hastings algorithm is introduced, for the purpose of sampling from a target distribution with strongly nonlinear support. The algorithm embeds the trajectory of the Markov chain into a reproducing kernel Hilbert space (RKHS), such that the feature space covariance of the samples informs the choice of proposal. The procedure is computationally efficient and straightforward to implement, since the RKHS moves can be integrated out analytically: our proposal distribution in the original space is a normal distribution whose mean and covariance depend on where the current sample lies in the support of the target distribution, and adapts to its local covariance structure. Furthermore, the procedure requires neither gradients nor any other higher order information about the target, making it particularly attractive for contexts such as Pseudo-Marginal MCMC. Kernel Adaptive Metropolis-Hastings outperforms competing fixed and adaptive samplers on multivariate, highly nonlinear target distributions, arising in both real-world and synthetic examples.
Paper link
Code link

Abstract
In biology, stochastic differential equations (SDEs) can be used to model the variability along time of biological processes (e.g. pharmacokinetics, neuron dynamics). In these SDEs, some components may be hidden. For example, bi-dimensional SDEs with only the first coordinate observed; SDEs with unobserved random parameters modeling the inter-subject variability. Hidden components give rise to a difficult estimation problem, the likelihood being not explicit. In this talk, we will survey how MCMC can be used to estimate parameters of SDEs, when some of the components are unobserved. Especially data-augmentation approaches and particle filters will be presented in that context.

Abstract
Originating in games, MCTS is a revolution in games and some forms of planning. It is an adaptive Monte Carlo sampling, in which simulated decisions get better. We will present an overview of the methods, mathematics and open problems.

Abstract
Drawing samples from a multivariate Gaussian distribution in high dimensions is a major step in many statistical inference problems, such as those encountered in machine learning, image processing and inverse problems. The particularity of the situation that we consider here is that the Gaussian distribution is defined by its precision matrix (inverse of the covariance) and the product of this precision matrix with the Gaussian mean. Direct sampling of a multivariate Gaussian distribution is traditionally performed using a Cholesky factorization of the covariance matrix. An alternative method is based on the resolution of a linear system involving the precision matrix, its Cholesky factor or square root. However, due to their numerical complexity and memory requirement, both approaches are infeasible in high dimensions, unless the precision matrix presents some specific structure. We describe is this talk a method that uses the reversible jump Monte Carlo Markov Chain framework to derive a Gaussian sampler (called RJPO, reversible jump perturbation optimization) incorporating an approximate resolution of a linear system with an accept-reject step allowing to ensure convergence. We also present an unsupervised adaptive scaling of the RJPO sampler with the aim to optimize the numerical efficiency in terms of minimal computation cost per effective sample.
 
Joint work with Jerome Idier and Clement Gilavert.

Abstract
In some application of Metropolis-Hastings algorithms, the exact computation of the acceptance ratio is too slow. 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 arising from, for example, Gibbs random fields. A natural approach in these cases is to replace the ratio by an approximation. The convergence of a Markov Chain under a perturbed version of the kernel is usually refered as the "stability of Markov Chains" theory. In a first time, I will introduce briefly introduce some results ensuring the stability of a Markov Chain. I will then focus on the application to big data. In this case, an efficient algorithm was proposed by: Korattikara, Chen et Welling (ICML, 2014). We will study its convergence through the tools of Bardenet, Doucet, & Holmes (ICML 2014). I will also focus on the Gibbs random fields example following our recent paper with N. Friel, A. Boland (UCD Dublin) et R. Everitt (University of Reading), arXiv:1403.5496.