This paper addresses the problem of statistical inference for latent continuous-time stochastic processes, which is often intractable, particularly for discrete state space processes described by Markov jump processes. To overcome this issue, we propose a new tractable inference scheme based on an entropic matching framework that can be embedded into the well-known expectation propagation algorithm. We demonstrate the effectiveness of our method by providing closed-form results for a simple family of approximate distributions and apply it to the general class of chemical reaction networks, which are a crucial tool for modeling in systems biology. Moreover, we derive closed form expressions for point estimation of the underlying parameters using an approximate expectation maximization procedure. We evaluate the performance of our method on various chemical reaction network instantiations, including a stochastic Lotka-Voltera example, and discuss its limitations and potential for future improvements. Our proposed approach provides a promising direction for addressing complex continuous-time Bayesian inference problems.

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