Estimates for hidden Markov models and particle approximations: Application to simultaneous mapping and localization.

Summary In this thesis, we are interested in the estimation of parameters in hidden Markov chains. We first consider the problem of online estimation (without saving observations) in the maximum likelihood sense. We propose a new method based on the Expectation Maximization algorithm called Block Online Expectation Maximization (BOEM). This algorithm is defined for hidden Markov chains with general state space and observation space. In the case of general state spaces, the BOEM algorithm requires the introduction of sequential Monte Carlo methods to approximate expectations under smoothing laws. The convergence of the algorithm then requires a control of the norm Lp of the Monte Carlo approximation error explicit in the number of observations and particles. A second part of this thesis is devoted to obtaining such controls for several sequential Monte Carlo methods. Finally, we study applications of the BOEM algorithm to simultaneous mapping and localization problems. The last part of this thesis is related to nonparametric estimation in hidden Markov chains. The problem considered is addressed in a specific framework. We assume that (Xk) is a random walk whose law of increments is known to within a scale factor a. We assume that, for any k, Yk is an observation of f(Xk) in an additive Gaussian noise, where f is a function we seek to estimate. We establish the identifiability of the statistical model and propose an estimate of f and a from the pairwise likelihood of the observations.