  Particle filtering (PF) is an often used method to estimate the states of
dynamical systems. A major limitation of the standard PF method is that the
dimensionality of the state space increases as the time proceeds and eventually
may cause degeneracy of the algorithm. A possible approach to alleviate the
degeneracy issue is to compute the marginal posterior distribution at each time
step, which leads to the so-called marginal PF method. A key issue in the
marginal PF method is to construct a good sampling distribution in the marginal
space. When the posterior distribution is close to Gaussian, the Ensemble
Kalman filter (EnKF) method can usually provide a good sampling distribution;
however the EnKF approximation may fail completely when the posterior is
strongly non-Gaussian. In this work we propose a defensive marginal PF (DMPF)
algorithm which constructs a sampling distribution in the marginal space by
combining the standard PF and the EnKF approximation using a multiple
importance sampling (MIS) scheme. An important feature of the proposed
algorithm is that it can automatically adjust the relative weight of the PF and
the EnKF components in the MIS scheme in each step, according to how
non-Gaussian the posterior is. With numerical examples we demonstrate that the
proposed method can perform well regardless of whether the posteriors can be
well approximated by Gaussian.
