Exposé de:

Alain Durmus (CMLA, ENS Paris-Salcay)

“Quantitative convergence of Unadjusted Langevin Monte Carlo and application to stochastic approximation”

Abstract : Stochastic approximation methods play a central role in maximum likelihood estimation problems involving intractable likelihood functions, such as marginal likelihoods arising in problems with missing or incomplete data, and in parametric empirical Bayesian estimation. Combined with Markov chain Monte Carlo algorithms, these stochastic optimisation methods have been successfully applied to a wide range of problems in science and industry. However, this strategy scales poorly to large problems because of methodological and theoretical difficulties related to using high-dimensional Markov chain Monte Carlo algorithms within a stochastic approximation scheme. This paper proposes to address these difficulties by using unadjusted Langevin algorithms to construct the stochastic approximation. This leads to a highly efficient stochastic optimisation methodology with favourable convergence properties that can be quantified explicitly and easily checked. The proposed methodology is demonstrated with three experiments, including a challenging application to high-dimensional statistical audio analysis and a sparse Bayesian logistic regression with random effects problem.

Exposé de:

Christian-Yann Robert (ENSAE)

*Testing for changes in the tail behaviour of Brown-Resnick Pareto processes*

Abstract. We consider the class of Pareto processes defined on [0,1] whose max-stable counterparts are Brown-Resnick processes. The aim of this paper is to propose a test whether the extreme value index function of a Pareto process of this class remains constant over [0,1]. We assume that we observe several independent Pareto processes over equispaced time grids of [0,1] but with possibly heterogeneous frequencies. We build a test based on the normalized total variations of the paths of these processes. We provide its properties under infill asymptotics and assess its finite sample performance through simulation experiments. We discuss how to proceed for random processes that belong to the domain of attraction of Brown-Resnick Pareto processes and illustrate the approach with an application to wind speed data from Germany.