Identifying Markov chain models from time-to-event data: an algebraic approach

TL;DR

This paper presents an algebraic method to identify transition rates in finite-state Markov chain models from phase-type distributions, with applications to biological data and solutions for models with identical distributions.

Contribution

It introduces a recursive and symbolic algebraic approach to solve the inverse problem of Markov model identification from phase-type data, including models with identical distributions.

Findings

Unique solutions for transition rates in certain Markov models

Recursive symbolic solutions applicable to any number of states

Ability to distinguish models with identical phase-type distributions

Abstract

Many biological and medical questions can be modeled using time-to-event data in finite-state Markov chains, with the phase-type distribution describing intervals between events. We solve the inverse problem: given a phase-type distribution, can we identify the transition rate parameters of the underlying Markov chain? For a specific class of solvable Markov models, we show this problem has a unique solution up to finite symmetry transformations, and we outline a recursive method for computing symbolic solutions for these models across any number of states. Using the Thomas decomposition technique from computer algebra, we further provide symbolic solutions for any model. Interestingly, different models with the same state count but distinct transition graphs can yield identical phase-type distributions. To distinguish among these, we propose additional properties beyond just the time to the next event. We demonstrate the method's applicability by inferring transcriptional regulation models from single-cell transcription imaging data.