Séance du 21 septembre 2020

Séance organisée par Estelle Kuhn et Marie-Luce Taupin.

Lieu : IHP, amphi Hermite

14.00 : Guillaume Kon Kam King (Université Paris-Saclay, INRAE)

Titre : Exact inference for a class of non-linear hidden Markov models on general state spaces

Résumé : Filtering hidden Markov models, or sequential Bayesian inference on the hidden state of a signal, is analytically tractable only for a handful of models. Examples are finite-dimensional state space models and linear Gaussian systems (Baum-Welch and Kalman filters). Recently, Papaspiliopoulos et al. ([1], [2])⁠ proposed a principled approach for extending the realm of analytically tractable models, exploiting a duality relation between the hidden process and an auxiliary process. Then, the solution of the filtering problem consists in a finite mixture of distributions. We study the computational effort required to implement this strategy for two parametric and nonparametric models: the Cox-Ingersoll-Ross process, the K-dimensional Wright-Fisher process, the Dawson-Watanabe process and the Fleming-Viot process. In all cases, the number of components involved in the filtering distributions increases rapidly with the number of observations. Although this could render the algorithm impractical for long observation sequences and undermine its practical relevance, the mathematical form of the filtering distributions suggest that the number of components which contribute most to the mixture remains small. This suggests several efficient natural approximation strategies. We assess the performance of these strategies in terms of accuracy, speed and prediction, benchmarked against the exact solution.

A preprint of this work is available at: https://arxiv.org/abs/2006.03452

[1] O. Papaspiliopoulos and M. Ruggiero, “Optimal filtering and the dual process,” Bernoulli, vol. 20, no. 4, pp. 1999–2019, 2014.

[2] O. Papaspiliopoulos, M. Ruggiero, and D. Spano, “Conjugacy properties of time-evolving Dirichlet and gamma random measures,” Electron. J. Stat., vol. 10, no. 2, pp. 3452–3489, 2016.

15.00 : Chiara Amorino (Université d'Evry)

Titre : Minimax rate of estimation for the stationary distribution of jump-processes over anisotropic Holder classes.

Résumé : We consider the solution X = (Xt)t>0 of a multivariate stochastic differential equation with Levy-type jumps and with unique invariant probability measure with density pi. We assume that a continuous record of observations XT = (Xt)0<t<T is available. In the case without jumps, Dalalyan and Reiss [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 achieve convergence rates faster than the ones found by Dalalyan and Reiss [1] and Strauch [3] for d>=3 and a rate which depends on the degree of the jumps in the one-dimensional setting. We also 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 pi of the jump diffusion X. Moreover, we obtain a minimax lower bound on the L2-risk for pointwise estimation, with the same rate up to a log(T) term. It implies that, on a class of diffusions whose invariant density belongs to the anisotropic Holder class we are considering, it is impossible to find an estimator with a rate of estimation faster than the one we propose.

Joint with Arnaud Gloter

[1] Dalalyan, A. and Reiss, M. (2007). Asymptotic statistical equivalence for ergodic diffusions: 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 : Céline Duval (Université Paris-Descartes)

Titre : Statistics for Gaussian Random Fields with Unknown Location and Scale using Lipschitz-Killing Curvatures

Résumé : We study three geometrical characteristics for the excursion sets of a 2-dimensional standard (centered and unit variance) stationary isotropic random field X. These characteristics can be estimated without bias if the field satisfies a kinematic formula, such as a smooth Gaussian field or some shot noise fields. If the field is Gaussian, we show how to remove the constraining assumption that the field is standard and adapt the previous estimators. We illustrate how these quantities can be used to recover some parameters of X and perform testing procedures. Finally, we use these tools to build a test to determine if two images of excursion sets can be compared. This test is applied on both synthesized and real mammograms.