The problem of estimating the state of a distributed finite-state Markov process consisting of several interacting finite-state systems each of whose transition probabilities are influenced by the states of the other processes is examined. The observations on which the estimation procedure is based are continuous signals containing signatures indicative of the occurrence of particular events in the various finite-state systems. The problem of electrocardiogram analysis serves both as the primary motivation for this investigation and as the source of a case study we describe. The principal focus of the paper is on the development of an approach that overcomes the combinatorial explosion of truly optimal estimation algorithms. We accomplish this by constructing a systematic design methodology in which the resulting estimator consists of several interacting estimators, each focusing on a particular sub-process, Important questions that we address concern the way in which these estimators interact and the method each estimator uses to account for the influence of other sub-processes in its own model.