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

arXiv: https://arxiv.org/abs/2112.10971 · HTML: https://mlsanity.com/paper/2112.10971/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 42 references that are in the map.
- First posted on arXiv 21 Dec 2021.

## Builds on (most cited first, 5 of 5)

- [Rank-normalization, folding, and localization: An improved widehatR for assessing convergence of MCMC](https://mlsanity.com/paper/1903.08008/) (2019)
- [The Spacey Random Walk: a Stochastic Process for Higher-order Data](https://mlsanity.com/paper/1602.02102/) (2016)
- [Direct statistical inference for finite Markov jump processes via the matrix exponential](https://mlsanity.com/paper/1809.07110/) (2018)
- [DD-αAMG on QPACE 3](https://mlsanity.com/paper/1710.07041/) (2017)
- [Grid on QPACE 4](https://mlsanity.com/paper/2112.01852/) (2021)

## Abstract (arXiv)

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 θ. Computing the probability distribution over states at time t requires the matrix exponential exp(tQ), and inferring θ from data requires its derivative partialexp!(tQ)/partialθ. 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 partialexp!(tQ)/partialθ, 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.