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 $1/\sqrt{N}$ in terms of the sample size $N$, 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  [30] . This preliminary result has been extended to a non–linear / non–Gaussian model, using the Laplace method [23] . 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 $N$ goes to infinity, and to analyze its dependence on the dimension.