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Approximate solutions of the chemical master equation and the chemicalFokker-Planck equation are an important tool in the analysis of biomolecularreaction networks. Previous studies have highlighted a number of problems withthe moment-closure approach used to obtain such approximations, calling it anad-hoc method. In this article, we give a new variational derivation ofmoment-closure equations which provides us with an intuitive understanding oftheir properties and failure modes and allows us to correct some of theseproblems. We use mixtures of product-Poisson distributions to obtain a flexibleparametric family which solves the commonly observed problem of divergences atlow system sizes. We also extend the recently introduced entropic matchingapproach to arbitrary ansatz distributions and Markov processes, demonstratingthat it is a special case of variational moment closure. This provides us witha particularly principled approximation method. Finally, we extend the aboveapproaches to cover the approximation of multi-time joint distributions,resulting in a viable alternative to process-level approximations which areoften intractable.

A variational approach to moment-closure approximations for the kinetics of biomolecular reaction networks