Maximum likelihood estimation for nonembeddable Markov chains when the cycle length is shorter than the data observation interval

TL;DR

This paper investigates the challenges of maximum likelihood estimation for Markov chains when data are collected at intervals longer than the chain's cycle length, highlighting issues with root extraction and non-convex optimization.

Contribution

It demonstrates the difficulty of estimating the transition matrix in such cases and explores local convergence issues through case studies, suggesting the need for alternative methods.

Findings

Maximum likelihood estimation becomes complex when data intervals exceed cycle length.

The $T$th root of the estimated transition matrix may not be valid or unique.

Global maximum likelihood estimates are hard to find as model complexity increases.

Abstract

Time-homogeneous Markov chains are often used as disease progression models in studies of cost-effectiveness and optimal decision-making. Maximum likelihood estimation of these models can be challenging when data are collected at a time interval longer than the model's transition cycle length. For example, it may be necessary to estimate a monthly transition model from data collected annually. The likelihood for a time-homogeneous Markov chain with transition matrix $\mathbf{P}$ and data observed at intervals of $T$ cycles is a function of $\mathbf{P}^T.$ The maximum likelihood estimate of $\mathbf{P}^T$ is easily obtained from the data. The $T$th root of this estimate would then be a maximum likelihood estimate for $\mathbf{P}.$ However, the $T$th root of $\mathbf{P}^T$ is not necessarily a valid transition matrix. Maximum likelihood estimation of $\mathbf{P}$ is a constrained optimization problem when a valid root is unavailable. The optimization problem is not convex. Local convergence is explored in several case studies through graphical representations of a grid search. The example cases use disease progression data from the literature as well as synthetic data. The global maximum likelihood estimate is increasingly difficult to locate as the number of cycles or the number of states increases. What seems like a straightforward estimation problem can be challenging even for relatively simple models. Researchers should consider alternatives to likelihood maximization or alternative models.