Bayesian Estimation and Tracking: A Practical Guide

5.2 Derivation of the Kalman Filter Correction (Update) Equations Revisited

In this section, we provide an alternate derivation of the Kalman filter correction equations (3.43), (3.44), and (3.47), based on the assumption that all conditional densities are Gaussian. Bayes' law provides a link between the posterior density and the joint density of x_n with z_n resulting in

(5.10)

Now defining the joint vector

(5.11)

(5.10) can be written as

(5.12)

The general form for a multivariate Gaussian distribution can be written as

(5.13)

Let all densities in (5.12) be Gaussian so that

(5.14)

(5.15)

(5.16)

Ignoring the factor of , the exponent of the exponential in the analytical expression for the ...