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 xn with zn resulting in
Now defining the joint vector
(5.10) can be written as
The general form for a multivariate Gaussian distribution can be written as
Let all densities in (5.12) be Gaussian so that
Ignoring the factor of , the exponent of the exponential in the analytical expression for the ...