Distinguishing between stochastic models of heterogeneity and contagion

Abstract

Historically, the Poisson process has been the “benchmark” model for many stochastic processes. When event counts from a particular process fail to be described adequately by a Poisson distribution, a researcher may turn to generalized Poisson processes to model the empirical data more accurately. Two common generalizations are the (1) heterogeneous Poisson process, in which the process rate is a Gamma random variable, and (2) contagious Poisson process, in which the process rate depends linearly on the current state of the process. Paradoxically, both the heterogeneous and contagious processes yield the same theoretical distribution for the number of events that occur in a single interval of time. Consequently, distinguishing between these two using event counts can be difficult. This situation is discussed, first reviewing the models and distribution, then giving strategies for choosing between them using event count data, and demonstrating these techniques on episodes of hospitalization for schizophrenia.