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

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, $${\mathsf {Q}}$$ Q , since $$\exp ({\mathsf {Q}}t)$$ exp ( 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 $${\mathsf {Q}}$$ 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.

PhaseTypeR: phase-type distributions in R with reward transformations and a view towards population genetics

“…supplementary material in Zeng, Charlesworth, and Hobolth (2021)). Recent progress for calculating the matrix exponential for large rate matrices is available in Sherlock (2021).…”

Exact Bayesian Inference for Discretely Observed Markov Jump Processes Using Finite Rate Matrices

“…Thus, e M can be obtained can be obtained from e M/2 s by squaring s times. We calculate e ρP/2 s using the uniformisation method (but without the ν term, as in Reibman and Trivedi, 1988;Pulungan and Hermanns, 2018), and revert to vector-matrix multiplications before the final squaring; see Sherlock (2021) for further details.…”

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

“…is smaller than a preset tolerance ε. The required number m of iterations is in O(γ) (Reibman and Trivedi 1988) and can be determined, e.g., using the numerically robust method by Sherlock (2021).…”