Particle approximation and the Laplace method for Bayesian filtering

Overview & Synopsis

The thesis deals with the contribution of the Laplace method to the approximation of the Bayesian filter in hidden Markov models with continuous state--space, i.e. in a sequential framework, with target tracking as the main application domain. Originally, the Laplace method is an asymptotic method used to compute integrals, i.e. in a static framework, valid in theory as soon as the function to be integrated exhibits an increasingly dominating maximum point, which brings the essential contribution to the integral. The two main contributions of the thesis are the following. Firstly, we have combined the Laplace method and particle filters: indeed, it is well-known that sequential Monte Carlo methods based on importance sampling are inefficient when the weighting function (here, the likelihood function) is too much spatially localized, e.g. when the variance of the observation noise is too small, whereas this is precisely the situation where the Laplace method is efficient and theoretically justified, hence the natural idea of combining the two approaches. We thus propose an algorithm associating the Laplace method and particle filtering, called the Laplace particle filter. Secondly, we have analyzed the approximation of the Bayesian filter based on the Laplace method only (i.e. without any generation of random samples): the objective has been to control the propagation of the approximation error from one time step to the next time step, in an appropriate asymptotic framework, e.g. when the variance of the observation noise goes to zero, or when the variances of the model noise and of the observation noise jointly go (with the same rate) to zero, or more generally when the information contained in the system goes to infinity, with an interpretation in terms of identifiability.