We propose a novel Markov chain Monte-Carlo (MCMC) method for reverse engineering the topological structure of stochastic reaction networks, a notoriously challenging problem that is relevant in many modern areas of research, like discovering gene regulatory networks or analyzing epidemic spread. The method relies on projecting the original time series trajectories onto information rich summary statistics and constructing the appropriate synthetic likelihood function to estimate reaction rates. The resulting estimates are consistent in the large volume limit and are obtained without employing complicated tuning strategies and expensive resampling as typically used by likelihood-free MCMC and approximate Bayesian methods. To illustrate run time improvements that can be achieved with our approach, we present a simulation study on inferring rates in a stochastic dynamical system arising from a density dependent Markov jump process. We then apply the method to two real data examples: the RNA-seq data from zebrafish experiment and the incidence data from 1665 plague outbreak at Eyam, England.
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