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Biochemical reaction networks are an amalgamation of reactions where eachreaction represents the interaction of different species. Generally, thesenetworks exhibit a multi-scale behavior caused by the high variability inreaction rates and abundances of species. The so-called jump-diffusionapproximation is a valuable tool in the modeling of such systems. Theapproximation is constructed by partitioning the reaction network into a fastand slow subgroup of fast and slow reactions, respectively. This enables themodeling of the dynamics using a Langevin equation for the fast group, while aMarkov jump process model is kept for the dynamics of the slow group. Mostoften biochemical processes are poorly characterized in terms of parameters andpopulation states. As a result of this, methods for estimating hiddenquantities are of significant interest. In this paper, we develop a tractableBayesian inference algorithm based on Markov chain Monte Carlo. The presentedblocked Gibbs particle smoothing algorithm utilizes a sequential Monte Carlomethod to estimate the latent states and performs distinct Gibbs steps for theparameters of a biochemical reaction network, by exploiting a jump-diffusionapproximation model. The presented blocked Gibbs sampler is based on the twodistinct steps of state inference and parameter inference. We estimate statesvia a continuous-time forward-filtering backward-smoothing procedure in thestate inference step. By utilizing bootstrap particle filtering within abackward-smoothing procedure, we sample a smoothing trajectory. For estimatingthe hidden parameters, we utilize a separate Markov chain Monte Carlo samplerwithin the Gibbs sampler that uses the path-wise continuous-time representationof the reaction counters. Finally, the algorithm is numerically evaluated for apartially observed multi-scale birth-death process example.

Bayesian Inference for Jump-Diffusion Approximations of Biochemical Reaction Networks