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.

给定时间齐次、有限状态空间马尔可夫链的噪声、部分观察，理论上可以通过其速率矩阵或无穷小生成器\({\mathsf {Q}}\)获得概念上简单的直接统计推断，因为\(\exp ({\mathsf {Q}}t)\)是时间t上的转移矩阵。然而，可能是因为统计学家常用的编程语言中矩阵求幂工具不足，或者认为必要的计算过于昂贵，具有大但有限状态空间的连续时间马尔可夫链的统计推断通常通过粒子 MCMC 或其他相对复杂的推理方案。当，在许多应用程序中\({\mathsf {Q}}\)来自反应网络，它通常是稀疏的。我们描述了已知算法的变体，这些算法允许快速、稳健和准确地评估非负向量与大型稀疏速率矩阵的指数的乘积。我们的实现使用了最近开发的、高效的线性代数工具，这些工具利用了这种稀疏性。我们展示了关键算法在种群中两个等位基因混合模型和易感-感染-去除流行病模型上的直接统计应用。