Discrete-time, discrete-state multistate Markov models from the perspective of algebraic statistics

TL;DR

This paper explores the algebraic structure of discrete-time, discrete-state multistate Markov models, establishing a connection with algebraic statistics, describing their polynomial relations, and analyzing maximum likelihood estimation.

Contribution

It provides a comprehensive algebraic characterization of multistate Markov models, including their vanishing ideals and polynomial relations, and compares homogeneous and nonhomogeneous cases.

Findings

Complete description of vanishing ideals as toric ideals for nonhomogeneous models

Polynomial relations differ between homogeneous and nonhomogeneous models

Algebraic formulas for maximum likelihood estimation are validated with data applications

Abstract

We study discrete-time, discrete-state multistate Markov models from the perspective of algebraic statistics. These models are widely studied in event history analysis, and are characterized by the state space, the initial distribution and the transition probabilities. A finite path under the multistate Markov model is a particular set of states occupied at finite time instances $\{1, \dots, n\}$. The main goal of this paper is to establish a bridge between event history analysis and algebraic statistics. The joint probabilities of finite paths in these models have a natural monomial parametrization in terms of the initial distribution and the transition probabilities. We study the polynomial relations among joint path probabilities. When the statistical constraints on the parameters are disregarded, nonhomogeneous multistate Markov models of arbitrary order can be viewed as slices of decomposable hierarchical models. This yields a complete description of their vanishing ideals as toric ideals generated by explicit families of binomials. Moreover, the variety of this vanishing ideal equals the nonhomogeneous multistate Markov model on the probability simplex. In contrast, homogeneous multistate Markov models exhibit different algebraic behavior, as time homogeneity imposes additional polynomial relations, leading to vanishing ideals that are strictly larger than in the nonhomogeneous case. We also derive families of binomial relations that vanish on homogeneous multistate Markov models. We investigate maximum likelihood estimation from statistical and algebraic perspectives. For nonhomogeneous models, classical and algebraic formulas agree; in the homogeneous case, the algebraic approach is more complex. Lastly, we provide data applications where we demonstrate the statistical theory to obtain the maximum likelihood estimates of the parameters under specific multistate Markov models.