  Given noisy, partial observations of a time-homogeneous, finite-statespace
Markov chain, conceptually simple, direct statistical inference is available,
in theory, via its rate matrix, or infinitesimal generator, $\mathsf{Q}$, since
$\exp (\mathsf{Q}t)$ is the transition matrix over time $t$. However, perhaps
because of inadequate tools for matrix exponentiation in programming languages
commonly used amongst statisticians or a belief that the necessary calculations
are prohibitively expensive, statistical inference for continuous-time Markov
chains with a large but finite state space is typically conducted via particle
MCMC or other relatively complex inference schemes.
  When, as in many applications $\mathsf{Q}$ arises from a reaction network, it
is usually sparse. We describe variations on known algorithms which allow fast,
robust and accurate evaluation of the product of a non-negative vector with the
exponential of a large, sparse rate matrix. Our implementation uses relatively
recently developed, efficient, linear algebra tools that take advantage of such
sparsity. We demonstrate the straightforward statistical application of the key
algorithm on a model for the mixing of two alleles in a population and on the
Susceptible-Infectious-Removed epidemic model.
