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Given noisy, partial observations of a time-homogeneous, finite-statespaceMarkov 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, perhapsbecause of inadequate tools for matrix exponentiation in programming languagescommonly used amongst statisticians or a belief that the necessary calculationsare prohibitively expensive, statistical inference for continuous-time Markovchains with a large but finite state space is typically conducted via particleMCMC or other relatively complex inference schemes.  When, as in many applications $\mathsf{Q}$ arises from a reaction network, itis usually sparse. We describe variations on known algorithms which allow fast,robust and accurate evaluation of the product of a non-negative vector with theexponential of a large, sparse rate matrix. Our implementation uses relativelyrecently developed, efficient, linear algebra tools that take advantage of suchsparsity. We demonstrate the straightforward statistical application of the keyalgorithm on a model for the mixing of two alleles in a population and on theSusceptible-Infectious-Removed epidemic model.

Direct statistical inference for finite Markov jump processes via the matrix exponential