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

TL;DR

This paper introduces efficient algorithms for direct statistical inference of finite-state continuous-time Markov chains using matrix exponential computations, especially leveraging sparsity in rate matrices for improved accuracy and speed.

Contribution

It presents novel variations of existing algorithms that enable fast, robust, and accurate evaluation of matrix exponentials for large, sparse rate matrices in Markov processes.

Findings

Algorithms perform well on population genetics models

Efficient handling of large, sparse matrices demonstrated

Facilitates direct inference without complex MCMC methods

Abstract

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.