  Markov jump process models have many applications across science. Often,
these models are defined on a state-space of product form and only one of the
components of the process is of direct interest. In this paper, we extend the
marginal process framework, which provides a marginal description of the
component of interest, to the case of fully coupled processes. We use entropic
matching to obtain a finite-dimensional approximation of the filtering
equation, which governs the transition rates of the marginal process. The
resulting equations can be seen as a combination of two projection operations
applied to the full master equation, so that we obtain a principled model
reduction framework. We demonstrate the resulting reduced description on the
totally asymmetric exclusion process. An important class of Markov jump
processes are stochastic reaction networks, which have applications in chemical
and biomolecular kinetics, ecological models and models of social networks. We
obtain a particularly simple instantiation of the marginal process framework
for mass-action systems by using product-Poisson distributions for the
approximate solution of the filtering equation. We investigate the resulting
approximate marginal process analytically and numerically.
