On the duality for nonlinear filtering

In this talk, I will present the dual control system for the hidden Markov model (HMM). This is a direct extension of classical duality between observability and controllability of linear systems. The classic relationship is extended to the linear-Gaussian filtering problem, where the minimum variance estimation problem is transformed into an optimal regulator problem on the dual system. The dual formulation is extensively used to prove the stability of the Kalman-Bucy filter.

In the first part of the talk, I review the duality in linear systems theory, and then I will revisit different types of variational formulations for nonlinear filtering in literature. Next, I will present our recent works on the dual control system and the optimal control formulation of the nonlinear filtering problem. The dual system is given by a backward stochastic differential equation (BSDE), and the optimal control objective consists of an energy term and quadratic control cost. In the last part, I discuss utility of this formulation: (1) analysis on the stability of the nonlinear filter against its initialization; (2) applications on approximation algorithms for the nonlinear filter.