Chapter 5
The Gaussian Noise Case: Multidimensional Integration of Gaussian-Weighted Distributions
At the end of Chapter 3, the Bayesian point estimation equations were developed using general probability density functions. In this part of the book, it will be assumed that the dynamic and observation equations constitute Gaussian processes. Under this assumption all of the distribution functions contained in the point estimator equations become Gaussian. It is well known that the first two moments of a Gaussian density characterize the density completely. Therefore, a recursive propagation of estimates of the first two moments produces an optimal estimation method for Gaussian processes. The subject of this part of the book includes the derivation of numerous methods for solving the density-weighted predictive point estimates developed in Chapter 3 for the special case of Gaussian densities.
In Section 3.4, a set of Kalman filter update equations were developed that levied no requirements on the linearity of the processes or the form of the densities associated with those processes. The main assumption made was that the posterior point estimate of xn, written as 
, be a linear function of the latest observation 
. Additional requirements on the point estimator included that the estimation ...
