Derivation of the Kalman filter in a Bayesian filtering perspective

Abstract

The Kalman filter is popularly known to be an optimal recursive implementation of the Bayesian prediction and correction in the sense that it minimises the estimated error covariance. The filter has been originally derived in this error minimising framework and there is extensive literature on the same. The Kalman filter has also been derived under other frameworks, like the maximum likelihood approach, etc., which all converge to the true posterior. In this paper we present a purely Bayesian filtering approach to the Kalman filter. We first build an analogy to the principles of Bayesian estimation and then present a step-by-step derivation for the Kalman filter following the Bayesian principles. From this derivation, we show that the Kalman filter gives a tractable solution to the Bayesian filtering process by computing the underlying probability densities exactly. This derivation is known to some in the research community but no formal article in the literature presents it in detail. This paper fills this gap and will be a good read for Bayesian enthusiasts. The filter is simulated in the proposed framework on a simple 4-D linear Gaussian model.