On optimal nonlinear estimation - Part II: Discrete observation

A general representation for the Joint conditional probability density of an arbitrary random signal process under discrete-time observation is obtained. This representation forms the cornerstone of the paper, and from it all other results are deduced. The conditional densities of prediction and smoothing are expressed in terms of filtering via the application of the general representation. The prediction and smoothing of a random process with linear dynamics and arbitrary a priori distribution are given to illustrate the applicability of the previous results in obtaining effectively computable formulas.