LOG-CONCAVE POSTERIOR DENSITIES ARISING IN CONTINUOUS FILTERING AND A MAXIMUM A POSTERIORI ALGORITHM.

Nonlinear filtering is fundamental to many engineering problems, as it involves inferring the state of a system given complicated dynamics and noisy observations. Some approaches to nonlinear filtering use the analysis of the underlying PDE or stochastic PDE governing the evolution of the posterior probability distribution over time, one approach, in particular, being the Yau-Yau method. In this paper, we give a maximum a posteriori (MAP) framework for the Yau-Yau method. Furthermore, we propose convex filtering, intermediate between linear and nonlinear filtering, which gives criteria under which the posterior preserves log-concavity. The key tool from the PDE is a result from Korevaar, giving criteria under which a quasilinear parabolic PDE preserves convexity. A bridge between the MAP estimator and the structure of the posterior is explained. Finally, we propose a novel numerical method based on iteration and apply this method to a tracking problem. [ABSTRACT FROM AUTHOR]