Bayes, models, and data

The Kalman filter is often described as a Bayesian processor. However, it is more than that. It is also a natural framework for introducing a physical model into an estimation scheme. As, such, it is a processor that can improve an estimation scheme in two distinct ways—by using prior statistics and by introducing a model. It is shown how the prior statistics are implicitly included in the Kalman update equation, and how models can be introduced in two places—the prediction equation and the measurement equation. Assuming linearity and Gaussianity, it is outlined how the Kalman equations evolve from the particle filter upon the assumptions of linearity and Gaussianity. Since the update equation for the particle filter is Bayes’ rule, it is clear that the resulting Kalman update equation is also a form of Bayes’ rule, thus verifying that the Kalman filter is indeed a Bayesian processor. It is then shown how a model can be introduced, further improving the quality of the estimate. An example is given, based ...