  Biochemical reaction networks are an amalgamation of reactions where each
reaction represents the interaction of different species. Generally, these
networks exhibit a multi-scale behavior caused by the high variability in
reaction rates and abundances of species. The so-called jump-diffusion
approximation is a valuable tool in the modeling of such systems. The
approximation is constructed by partitioning the reaction network into a fast
and slow subgroup of fast and slow reactions, respectively. This enables the
modeling of the dynamics using a Langevin equation for the fast group, while a
Markov jump process model is kept for the dynamics of the slow group. Most
often biochemical processes are poorly characterized in terms of parameters and
population states. As a result of this, methods for estimating hidden
quantities are of significant interest. In this paper, we develop a tractable
Bayesian inference algorithm based on Markov chain Monte Carlo. The presented
blocked Gibbs particle smoothing algorithm utilizes a sequential Monte Carlo
method to estimate the latent states and performs distinct Gibbs steps for the
parameters of a biochemical reaction network, by exploiting a jump-diffusion
approximation model. The presented blocked Gibbs sampler is based on the two
distinct steps of state inference and parameter inference. We estimate states
via a continuous-time forward-filtering backward-smoothing procedure in the
state inference step. By utilizing bootstrap particle filtering within a
backward-smoothing procedure, we sample a smoothing trajectory. For estimating
the hidden parameters, we utilize a separate Markov chain Monte Carlo sampler
within the Gibbs sampler that uses the path-wise continuous-time representation
of the reaction counters. Finally, the algorithm is numerically evaluated for a
partially observed multi-scale birth-death process example.
