Reduced Complexity HMM Filtering With Stochastic Dominance Bounds: A Convex Optimization Approach

Abstract

This paper uses stochastic dominance principles to construct upper and lower sample path bounds for Hidden Markov Model (HMM) filters. We consider an HMM consisting of an X-state Markov chain with transition matrix P. By using convex optimization methods for nuclear norm minimization with copositive constraints, we construct low rank stochastic matrices P and P̅ so that the optimal filters using P,P̅ provably lower and upper bound (with respect to a partially ordered set) the true filtered distribution at each time instant. Since P and P̅ are low rank (say R), the computational cost of evaluating the filtering bounds is O(X R) instead of O(X <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ). A Monte-Carlo importance sampling filter is presented that exploits these upper and lower bounds to estimate the optimal posterior. Finally, explicit bounds are given on the variational norm between the true posterior and the upper and lower bounds in terms of the Dobrushin coefficient.