Bayesian Estimation and Tracking: A Practical Guide

5.3 The General Bayesian Point Prediction Integrals for Gaussian Densities

Remembering the dynamic equation (5.1), and assuming that v_n is a zero-mean Gaussian random variable independent of both and it follows that

(5.32)

Similarly, in the observation equation (5.2), assume that w_n is a zero-mean Gaussian random variables independent of both x_n and z_n, that is

(5.33)

Now, from Bayes' law, the prior density can be obtained from

(5.34)

making Gaussian, since the product of two Gaussian densities is also Gaussian. Assuming that all density functions are Gaussian, we can identify

(5.35)

(5.36)

Assuming that v_n and w_n are zero-mean Gaussian processes, and substituting (5.36) into (5.3) and ...