Importance sampling applied to Pincus maximization for particle filter MAP estimation

Sequential Monte Carlo (SMC), or Particle Filters (PF), approximate the posterior distribution in nonlinear filtering arbitrarily well, but the problem how to compute a state estimate is not always straightforward. For multimodal posteriors, the maximum a posteriori (MAP) estimate is a logical choice, but it is not readily available from the SMC output. In principle, the MAP can be obtained by maximizing the posterior density obtained e.g. by the particle based approximation of the Chapman-Kolmogorov equation. However, this posterior is a mixture distribution with many local maxima, which makes the optimization problem very hard. We suggest an algorithm for estimating the MAP using the global optimization principle of Pincus and subsequently outline the frameworks for estimating the filter and marginal smoother MAP of a dynamical system from the SMC output.