Filtering for Random Finite Group Homomorphic Sequential Systems

Abstract

The nonlinear filtering problem has proven to be an extremely difficult one. When the filtering problem is described by a set of stochastic differential equations, the optimal nonlinear filter in general requires the solution of a stochastic partial differential equation for the conditional density or the solution of an infinite set of stochastic differential “moment” equations.¹ In the case of discrete-time partially observable finite-state Markov processes (POFSMP), the solution is conceptually simpler, as the conditional distribution can be computed sequentially via straightforward finite-dimensional difference equations.² However, even in this conceptually simple case, the nonlinear filtering problem can be computationally nontrivial. Specifically, we note that if we are considering an n-state POFSMP, a straightforward implementation of the conditional distribution update equations requires 0(n²) multiplications. For n of reasonable size this becomes an extremely demanding computational task.