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Exposés du vendredi 25 octobre 2019

### Résumés

1. Title: Geometric convergence and concentration inequalities for the Feynman-Kac genetic algorithm
By Xinyu Wang (Huazhong University of Science and Technology, Wuhan, China)
joint with Pierre Del Moral, Shulan Hu and Liming Wu.

Abstract. In this paper, we consider a genetic evolution model associated to a given Feynman-Kac flow (called also the simple genetic algorithm). We first obtain an estimate of the contraction coefficient of this interacting particle system in some suitable metric, independent of the number of particles in the system. Second, by transport-entropy inequality technique, we obtain some concentration inequalities for the particle system, uniform in time and in the number of particles.

2. Title: Maximum entropy estimator for Hidden Markov models: reduction to dimension 2
Shulan HU (University of Economics and Law, Wuhan, China. )
jonit with Xinyu Wang, Liming Wu.

Abstract. In this paper, we introduce the maximum entropy estimator based on 2-dimensional empirical distribution of the observation sequence of a Hidden Markov Model, when the sample size is big: in that case the maximum likelihood estimator (MLE) is too consuming in time by the classical Baum-Welch EM algorithm. We prove the consistency and the asymptotic normality of $\hat\theta^{ME}_n$ in a quite general framework, where the asymptotic covariance matrix is explicitly estimated in terms of the 2-dimensional Fisher information. To complement it we use also the 2-dimensional relative entropy to study the hypotheses testing problem. Furthermore we propose the gradient descent of 2-dimensional relative entropy algorithm for finding $\hat \theta_n^{ME}$, which works for very big $n$ and big number $m$ of the hidden states. Some numerical examples are furnished and commented for illustrating our theoretical results.