The development of an approach for obtaining statistical inferences about nonobservable processes that influence a processy(\cdot)which can be observed directly and which is assumed to be a mixture of continuous and discontinuous components is continued. The approach is based on probability-measure transformations and consists of finding the conditional probability of a nonobservable event in terms of the prior probability of that event and a functional of the observationsy(\cdot). The topics studied include optimal filtering, smoothing, and prediction estimates of the nonobservable process;M-ary hypothesis testing; performance lower-bounds; and stochastic control.

Optimum estimates of nonobservable random variables or random processes which influence the rate functions of a discrete time jump process (D) are derived. The approach used is based on the {\em a posteriori} probability of a nonobservable event expressed in terms of the {\em a priori} probability of that event and of the sample function probability of the DTJP. Thus a general representation is obtained for optimum estimates, and recursive equations are derived for minimum mean-squared error (MMSE) estimates. In general, MMSE estimates are nonlinear functions of the observations. The problem of estimating the rate of a DTJP when the rate is a random variable with a beta probability density function and the jump amplitudes are binomially distributed is considered. It is shown that the MMSE estimates ale linear. The class of beta density functions is rather rich and explains why there are insignificant differences between optimum unconstrained and linear MMSE estimates in a variety of problems.

The observation processyconsidered is an additive composition of continuous and discontinuous components. The additive Gaussian, point, and jump process models, treated separately in the past, are all included here simultaneously. Representations foryin terms of its innovations and following a Girsanov-type measure transformation are derived. These are then used to develop a measure form of Bayes' rule that provides a convenient tool for the study of estimation and decision problems arising in a variety of applications including communication and control.