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A class of regular jump processes (RJP's) is introduced. An RJP is described in terms of the intensity function of its associated stochastic point process and the state-transition density of its embedded random-state sequence. Expressions for the joint occurrence statistics of these processes are derived. Assuming that an information stochastic process causally modulates an observed RJP, we obtain the joint occurrence statistics of the resulting compound jump processes. We show the latter to incorporate appropriately the causal MMSE estimate of the conditional intensities and state-transition functions. The results are used to derive a general likelihood-ratio formula for information processing of RJP's. A separation is observed between the likelihood processor of the point process associated with the observed RJP and the processor associated with the embedded stochastic state sequence. Considering the detection of RJP's with uncertain (statistically known) probability measures, we obtain the optimal Bayes receiver as the appropriate compound likelihood processor and thus exhibit separation between the detection and filtering operations.