Séance du 16 mars 2020

Séance organisée par Estelle Kuhn et Marie-Luce Taupin.

Lieu : IHP, amphi Hermite

14.00 : Julien Stoehr (Université Paris-Dauphine)

Titre : Component-wise approximate Bayesian computation via Gibbs-like steps

Résumé : Approximate Bayesian computation methods are useful for generative models with intractable likelihoods. These methods are however sensitive to the dimension of the parameter space, requiring exponentially increasing resources as this dimension grows. To tackle this difficulty, we explore a Gibbs version of the ABC approach that runs component-wise approximate Bayesian computation steps aimed at the corresponding conditional posterior distributions, and based on summary statistics of reduced dimensions. While lacking the standard justifications for the Gibbs sampler, the resulting Markov chain is shown to converge in distribution under some partial independence conditions. The associated stationary distribution can further be shown to be close to the true posterior distribution and some hierarchical versions of the proposed mechanism enjoy a closed form limiting distribution. Experiments also demonstrate the gain in efficiency brought by the Gibbs version over the standard solution.

15.00 : Chiara Amorino (Université d'Evry - Paris Saclay)

Titre : Invariant adaptive density estimation for ergodic SDE with jumps over anisotropic classes.

Résumé : We consider the solution X = (X_t)t>0 of a multivariate stochastic differential equation with Levy-type jumps and with unique invariant probability measure with density mu. We assume that a continuous record of observations X^T = (X_t)0<t<T is available. In the case without jumps, Reiss and Dalalyan [1] and Strauch [3] have found convergence rates of invariant density estimators, under respectively isotropic and anisotropic Hölder smoothness constraints, which are considerably faster than those known from standard multivariate density estimation. We extend the previous works by obtaining, in presence of jumps, some estimators which have the same convergence rates they had in the case without jumps for d>=2 and a rate which depends on the degree of the jumps in the one-dimensional setting. We propose moreover a data driven bandwidth selection procedure based on the Goldenshluger and Lepski method [2] which leads us to an adaptive nonparametric kernel estimator of the stationary density mu of the jump diffusion X.

Joint with Arnaud Gloter

References :

[1] Dalalyan, A. and Reiss, M. (2007). Asymptotic statistical equivalence for ergodic di_usions: the multidimensional case. Probab. Theory Relat. Fields,137(1), 25{47.

[2] Goldenshluger, A., Lepski, O. (2011). Bandwidth selection in kernel density estimation: oracle inequalities and adaptive minimax optimality. The Annals of Statistics, 39(3), 1608-1632.

[3] Strauch, C. (2018). Adaptive invariant density estimation for ergodic diffusions over anisotropic classes. The Annals of Statistics, 46(6B), 3451-3480.

16.00 : Antoine Chambaz (Université Paris-Descartes)

Titre : Computationally fast targeted learning using adaptive survey sampling

Résumé : We address the practical construction of asymptotic confidence intervals (CIs) for smooth, real-valued statistical parameters by targeted learning from iid data in contexts where sample size is so large that it poses computational challenges. We observe some summary measure of all data and select a sub-sample from the complete data set by sampling with unequal inclusion probabilities based on the summary measures. Targeted learning is then carried out from the easier to handle sub-sample. We derive a central limit theorem for the targeted minimum loss estimator (TMLE) which enables the construction of the CIs. The inclusion probabilities can be optimized to reduce the asymptotic variance of the TMLE. We illustrate the procedure with an example where the parameter of interest is a variable importance measure of a continuous exposure on an outcome. We also conduct a simulation study and comment briefly on its results. This talk is based on joint works with P. Bertail, E. Joly and X. Mary

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