Bayesian adaptive filtering: Principles and practical approaches

While adaptive filtering is in principle intended for tracking non-stationary systems, most adaptive filtering algorithms have been designed for converging to a fixed unknown filter. When actually confronted with a non-stationary environment, they possess just one parameter (stepsize, forgetting factor) to adjust their tracking capability. Virtually the only existing optimal approach is the Kalman filter, in which the time-varying optimal filter is modeled as a vector AR(1) process. The Kalman filter is in practice never applied as an adaptive filter because of its complexity and large number of unknown parameters in its state-space (AR(1)) model. Here we consider optimal adaptive filtering for any stationary optimal filter evolution. We emphasize the various aspects of an optimal Bayesian approach, which not only include parameter variation bandwidth but also a priori parameter size and parameter dynamics. Finally we recommend some constrained versions of modest complexity and show how to estimate the parameters in the resulting Bayesian adaptive filters.