## Section: New Results

### Laplace and sequential Monte Carlo methods in Bayesian filtering

Participants : François Le Gland, Paul Bui--Quang.

This is a collaboration with Christian Musso (ONERA Palaiseau).

The Laplace method is a deterministic technique to approximate integrals,
and it has been widely used in Bayesian statistics, e.g. to compute
posterior means and variances [72] .
The approximation is consistent as the observations sample size goes to
infinity or as the observation noise intensity goes to zero, and the main
condition to apply the method is that the model should be identifiable.
The aim of [21] is to combine SMC methods
and the Laplace method in order to better approximate the posterior
density in nonlinear Bayesian filtering.
At each stage of the proposed algorithm,
a first approximate density is build from the current
population of particles, then an accurate estimate of the posterior mean
and covariance matrix is obtained using the Laplace method,
and these estimates are used to shift and rescale the population of
particles.
Overall, this procedure could be interpreted as another design of
an importance distribution that takes the observations into account.
The current work aims at using the Laplace method to cope with *weight
degeneracy* in particle filtering, a phenomenon that typically occurs
when the observation noise is small, which is precisely the situation
where the Laplace method is efficient.