Differentiated uniformization: A new method for inferring Markov chains on combinatorial state spaces including stochastic epidemic models

• URL: http://arxiv.org/abs/2112.10971v1
• Date: Tue, 21 Dec 2021 03:59:06 GMT
• Title: Differentiated uniformization: A new method for inferring Markov chains on combinatorial state spaces including stochastic epidemic models
• Authors: Kevin Rupp, Rudolf Schill, Jonas S\"uskind, Peter Georg, Maren Klever, Andreas L\"osch, Lars Grasedyck, Tilo Wettig, Rainer Spang
• Abstract summary: We provide an analogous algorithm for computing $partialexp!(tQ)theta$. We estimate monthly infection and recovery rates during the first wave of the COVID-19 pandemic in Austria.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: Motivation: 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 $\exp(tQ)$, and inferring $\theta$ from data requires its derivative $\partial\exp\!(tQ)/\partial\theta$. 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 $\exp(tQ)$ becomes feasible by the uniformization method, which does not require explicit storage of $Q$. Results: Here we provide an analogous algorithm for computing $\partial\exp\!(tQ)/\partial\theta$, 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. Availability: Implementation and data are available at https://github.com/spang-lab/TenSIR.

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