  In this work, we perform Bayesian inference tasks for the chemical master
equation in the tensor-train format. The tensor-train approximation has been
proven to be very efficient in representing high dimensional data arising from
the explicit representation of the chemical master equation solution. An
additional advantage of representing the probability mass function in the
tensor train format is that parametric dependency can be easily incorporated by
introducing a tensor product basis expansion in the parameter space. Time is
treated as an additional dimension of the tensor and a linear system is derived
to solve the chemical master equation in time. We exemplify the tensor-train
method by performing inference tasks such as smoothing and parameter inference
using the tensor-train framework. A very high compression ratio is observed for
storing the probability mass function of the solution. Since all linear algebra
operations are performed in the tensor-train format, a significant reduction of
the computational time is observed as well.
