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

arXiv: https://arxiv.org/abs/1809.07110 · HTML: https://mlsanity.com/paper/1809.07110/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Sep 2018.

## Built on by (most cited first, 2 of 2)

- [Differentiated uniformization: A new method for inferring Markov chains on combinatorial state spaces including stochastic epidemic models](https://mlsanity.com/paper/2112.10971/) (2021)
- [Neural Likelihood Approximation for Integer Valued Time Series Data](https://mlsanity.com/paper/2310.12544/) (2023)

## Abstract (arXiv)

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, mathsfQ, since exp (mathsfQt) 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 mathsfQ 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.