Comparison of various approaches for joint Wiener/Kalman filtering and parameter estimation with application to BASS

In recent years, the Kalman filter (KF) has encountered renewed interest, due to an increasing range of applications. Even though in many cases the state-space model may be linear, it is often only known up to the values of some parameters, usually related to the vector autoregressive process of the state evolution equation. In this paper, after finding motivation in some applications, we review a number of approaches for adaptive Kalman filtering (AKF), in which state and parameters get estimated jointly. We propose an improved version of the Extended KF (EKF) in which the estimation error covariance matrix is computed exactly assuming overall joint Gaussianity. We also compare the performance and Cramer Rao bounds (CRBs) of joint Maximum A Posteriori Maximum Likelihood (MAP-ML) estimation of Bayesian state and deterministic parameters, and marginalized ML estimation of the parameters, and relate this to the Expectation-Maximization KF (EM-KF). The perspectives involve also the Variational Bayesian KF (VB-KF).