Section: New Results
Impact of dimension in particle filtering and the Laplace method
Participants : François Le Gland, Paul Bui--Quang.
See 3.1 .
This is a collaboration with Christian Musso (ONERA Palaiseau).
Particle filtering is a widely used Monte Carlo method to approximate the posterior probability distribution in non–linear filtering, with an error scaling as in terms of the sample size , but otherwise independently of the underlying state dimension. However, it has been observed for a long time in practice that particle filtering can be quite inefficient when the dimension of the system is high. In a simple static linear Gaussian model, it has been possible indeed to check that the error on the estimation of the predicted likelihood, a quantitative indicator of the consistency between the prior distribution and the likelihood function, increases exponentially with the dimension  . This preliminary result has been extended to a non–linear / non–Gaussian model, using the Laplace method  . The Laplace method, which approximates multidimensional integrals accurately, has also been used to compute the asymptotic variance of the importance weights, as the sample size goes to infinity, and to analyze its dependence on the dimension.