Séance organisée par Judith Rousseau et Joseph Salmon.
Lieu : IHP, Amphithéâtre Perrin.
14h00 : Anne Sabourin (Telecom ParisTech, LTCI)
Titre : Dirichlet mixtures for multivariate extremes: model re-parametrization and Bayesian inference with censored data.
(travail en collaboration avec Philippe Naveau, Benjamin Renard et Anne-Laure Fougère)
Résumé : The dependence structure of multivariate extremes, defined as random vectors which jointly exceed large thresholds, can be characterized, up to marginal standardization, by an angular measure on the simplex, only subject to first moment constraints. Estimating the angular measure is thus, by nature, a non parametric problem. Finite Dirichlet mixtures can be used to approach weakly such angular measures but, in practice, the moment constraints make Bayesian inference very challenging in dimension greater than three.
We present a re-parametrization of the Dirichlet mixture model, in which the moment constraints are automatically satisfied. This allow for a natural prior specification as well as a simple implementation of a reversible-jump MCMC. Posterior consistency and ergodicity are verified. We illustrate the methods with a four-variate streamflow dataset, including historical information (the earliest flood has been recorded in 1604), which results in censored and missing data. Advantage is taken of the conditioning and marginalization properties of Dirichlet distributions to resolve censored likelihood issues within a data augmentation framework.
15h00 : Sylvain LeCorff (CRISM)
Titre : Continuous-time importance sampling for Jump diffusions.
Résumé : This talk introduces a new algorithm to sample from continuous-time jump diffusions and to estimate expectations of functionals of such diffusions. Recently, new exact algorithms have been proposed to draw samples from finite-dimensional distributions of diffusion processes and jump diffusions without any discretization step. These algorithms are based on a rejection sampling procedure and draw skeletons at some random time steps. However, these exact methods rely on strong assumptions such as the reducibility to a unit-volatility jump diffusion using the Lamperti transform. While this assumption can be proved easily for scalar diffusions, much stronger conditions are required in the multidimensional case.
In this contribution, we introduce a new algorithm to compute unbiased estimates of expectations of functionals of jump diffusions which can be used under weaker assumptions. This Jump Continuous Importance Sampling (JCIS) algorithm draws weighted skeletons using an importance sampling mechanism recently introduced for diffusion processes. In this case, the sampled paths are not distributed as the diffusion process but the weighted samples can be used to produce unbiased Monte Carlo estimates. The JCIS algorithm is compared to several other algorithms (Euler scheme with thinned jumps, Multilevel Monte Carlo path simulation, Jump Exact algorithm) using different models (Merton model, Sinus model, Double Jump model).
16h00 : Alexandre Tsybakov (CREST et Université Pierre et Marie Curie)
Titre : Comparaison des approches minimax en Statistique et Machine Learning.
Résumé : In the standard nonparametric regression setting, it is assumed that the model is well-specified, i.e., the unknown regression function belongs to a given functional class. The value of reference characterizing the best estimation is the minimax risk over this class. On the other hand, in Machine Learning no assumption is placed on the regression function; it does not necessarily belong to the specified class and the corresponding value of reference can be written as the minimax regret (minimax excess risk). The study of the minimax risk and of the minimax regret represents two parallel developments: the former has been analyzed mostly within Nonparametric Statistics, while the second -- within Statistical Learning Theory. This talk aims to bring out a connection between these two objects. We introduce a more general risk measure that realizes a smooth transition between the minimax risk and the minimax regret depending on the magnitude of the approximation error. The minimax risk and the minimax regret appear as the two extremities of this scale. The main result shows that, unless the functional class is extremely "massive", the minimax risk and minimax regret have the same asymptotic behavior. Furthermore, the optimal rates for the minimax regret and minimax risk are attained by one and the same procedure called the aggregation-of-leaders estimator while they are not attained by classical procedures such as empirical risk minimization and skeleton aggregation. For very "massive" classes the minimax risk turns out to be of smaller order than the minimax regret. As a by-product of general results, we get oracle inequalities for the excess risk involving Vapnik-Chervonenkis type classes and classes with polynomial growth of empirical entropy without the usual convexity assumption on the class. Finally, for a slightly modified method, we derive a bound on the excess risk of s-sparse convex aggregation improving that of Lounici (2007) and providing the optimal rate. Joint work with Alexander Rakhlin and Karthik Sridharan.