We consider continuous-time Markov chains that describe the stochastic evolution of a dynamical system by a transition-rate matrix Q which depends on a parameter \(\theta \). Computing the probability distribution over states at time t requires the matrix exponential \(\exp \,\left( tQ\right) \,\), and inferring \(\theta \) from data requires its derivative \(\partial \exp \,\left( tQ\right) \,/\partial \theta \). Both are challenging to compute when the state space and hence the size of Q is huge. This can happen when the state space consists of all combinations of the values of several interacting discrete variables. Often it is even impossible to store Q. However, when Q can be written as a sum of tensor products, computing \(\exp \,\left( tQ\right) \,\) becomes feasible by the uniformization method, which does not require explicit storage of Q. Here we provide an analogous algorithm for computing \(\partial \exp \,\left( tQ\right) \,/\partial \theta \), the differentiated uniformization method. We demonstrate our algorithm for the stochastic SIR model of epidemic spread, for which we show that Q can be written as a sum of tensor products. We estimate monthly infection and recovery rates during the first wave of the COVID-19 pandemic in Austria and quantify their uncertainty in a full Bayesian analysis. Implementation and data are available at https://github.com/spang-lab/TenSIR.

我们考虑连续时间马尔可夫链，它通过取决于参数\(\theta \) 的转移率矩阵Q来描述动力系统的随机演化。计算时间t状态的概率分布需要矩阵指数\(\exp \,\left( tQ\right) \,\)，从数据推断\(\theta \)需要其导数\(\partial \exp \,\left( tQ\right) \,/\partial \theta \)。当状态空间和Q的大小很大时，两者的计算都具有挑战性。当状态空间由多个相互作用的离散变量的值的所有组合组成时，就会发生这种情况。通常甚至不可能存储Q。然而，当Q可以写为张量积之和时，通过均匀化方法计算\(\exp \,\left( tQ\right) \,\)变得可行，不需要显式存储Q。这里我们提供了一个类似的计算\(\partial \exp \,\left( tQ\right) \,/\partial \theta \) 的算法，即微分均匀化方法。我们展示了流行病传播的随机 SIR 模型的算法，其中我们表明Q可以写成张量乘积之和。我们估计了奥地利第一波 COVID-19 大流行期间的每月感染率和康复率，并通过完整的贝叶斯分析量化了其不确定性。实现和数据可在 https://github.com/spang-lab/TenSIR 获取。