A Bayesian robust Kalman smoothing framework for state-space models with uncertain noise statistics

The classical Kalman smoother recursively estimates states over a finite time window using all observations in the window. In this paper, we assume that the parameters characterizing the second-order statistics of process and observation noise are unknown and propose an optimal Bayesian Kalman smoother (OBKS) to obtain smoothed estimates that are optimal relative to the posterior distribution of the unknown noise parameters. The method uses a Bayesian innovation process and a posterior-based Bayesian orthogonality principle. The optimal Bayesian Kalman smoother possesses the same forward-backward structure as that of the ordinary Kalman smoother with the ordinary noise statistics replaced by their effective counterparts. In the first step, the posterior effective noise statistics are computed. Then, using the obtained effective noise statistics, the optimal Bayesian Kalman filter is run in the forward direction over the window of observations. The Bayesian smoothed estimates are obtained in the backward step. We validate the performance of the proposed robust smoother in the target tracking and gene regulatory network inference problems.