Boolean Kalman filter and smoother under model uncertainty

Abstract

Partially-observed Boolean dynamical systems (POBDS) are a general class of nonlinear state-space models that provide a rich framework for modeling many complex dynamical systems. The model consists of a hidden Boolean state process, observed through an arbitrary noisy mapping to a measurement space. The optimal minimum mean-square error (MMSE) POBDS state estimators are the Boolean Kalman Filter and Smoother. However, in many practical problems, the system parameters are not fully known and must be estimated. In this paper, for POBDS under model uncertainty, we derive an optimal Bayesian estimator for state and parameter estimation. The exact algorithms are derived for the case of discrete and finite parameter space, and for general parameter spaces, an approximate Markov-Chain Monte-Carlo (MCMC) implementation is introduced. We demonstrate the performance of the proposed methodology by means of numerical experiments with POBDS models of gene regulatory networks observed through noisy measurements.