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

TL;DR

This work develops an algebraic-statistics framework for discrete-time, discrete-state multistate Markov models, expressing joint path probabilities with a monomial parametrization and studying their toric vanishing ideals. It shows nonhomogeneous $k$-th order models are coordinate slices of decomposable hierarchical models, with explicit binomial generators, while homogeneous models accrue additional polynomial constraints from time-homogeneity; this leads to distinct algebraic and statistical behaviors, especially for MLE. The paper analyzes maximum likelihood estimation from both statistical and algebraic perspectives, proving ML degree one for decomposable (nonhomogeneous) cases and illustrating complexities in the homogeneous case, supported by data applications to illness-death models and a Shakespeare word corpus. These results illuminate identifiability, model equivalence, and inference under structural constraints, and open questions about censoring, higher-order homogeneous models, and connections to graphical and colored-model frameworks.

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.

Figures (3)

• Figure 1: Simple survival model. An individual starts in state $0$ and remains in that state until transitioning to state $1$. For three discrete time points, the process admits three possible paths (or trajectories): $(0,0,0)$, $(0,0,1)$, and $(0,1,1)$. Once the process enters state $1$, it remains there; in this example, state $1$ is an absorbing state.
• Figure 2: An alternating 2-state process. An individual starts in state $0$ and remains there until transitioning to state $1$. From state $1$, the process may return to state $0$, and multiple transitions between the two states are possible. For three discrete time points, the process admits four possible paths (or trajectories): $(0,0,0)$, $(0,0,1)$, $(0,1,0)$, and $(0,1,1)$.
• Figure 3: An illness-death model. The initial state is $0$, and state $2$ is absorbing. For three discrete time points, the process admits six possible paths: $(0,0,0)$, $(0,0,1)$, $(0,1,1)$, $(0,0,2)$, $(0,1,2)$, and $(0,2,2)$.

Theorems & Definitions (25)

• Example 1: Illness-death model
• Example 2: Reversible illness-death model
• Definition 3.1
• Remark 3.2
• Definition 4.1
• Definition 4.2
• Lemma 4.3
• proof
• Proposition 4.4
• proof