Beyond time-homogeneity for continuous-time multistate Markov models

Multistate Markov models are a canonical parametric approach for data modeling of observed or latent stochastic processes supported on a finite state space. Continuous-time Markov processes describe data that are observed irregularly over time, as is often the case in longitudinal medical data, for example. Assuming that a continuous-time Markov process is time-homogeneous, a closed-form likelihood function can be derived from the Kolmogorov forward equations – a system of differential equations with a well-known matrix-exponential solution. Unfortunately, however, the forward equations do not admit an analytical solution for continuous-time, time-<i>inhomogeneous</i> Markov processes, and so researchers and practitioners often make the simplifying assumption that the process is piecewise time-homogeneous. In this paper, we provide intuitions and illustrations of the potential biases for parameter estimation that may ensue in the more realistic scenario that the piecewise-homogeneous assumption is violated, and we advocate for a solution for likelihood computation in a truly time-inhomogeneous fashion. Particular focus is afforded to the context of multistate Markov models that allow for state label misclassifications, which applies more broadly to hidden Markov models (HMMs), and Bayesian computations bypass the necessity for computationally demanding numerical gradient approximations for obtaining maximum likelihood estimates (MLEs). Supplemental materials are available online.

多状态马尔可夫模型（Multistate Markov Models）是一类经典的参数化建模方法，用于对有限状态空间上的观测或潜随机过程开展数据建模。连续时间马尔可夫过程可用于刻画随时间非规则观测得到的数据，例如纵向医疗数据中便常见此类场景。若假设连续时间马尔可夫过程满足时间齐次性，则可从柯尔莫哥洛夫前向方程（Kolmogorov Forward Equations）推导出闭式似然函数——该微分方程组存在广为人知的矩阵指数解。然而遗憾的是，对于连续时间非齐次（time-inhomogeneous）马尔可夫过程，前向方程并无解析解，因此研究者与从业者通常会做出过程为分段时间齐次的简化假设。本文针对分段齐次假设不成立的更符合实际的场景，阐述了参数估计中可能出现的潜在偏差，并给出直观解释与示例；同时提出了一种真正适配时间非齐次性的似然计算方案。本文特别聚焦于允许状态标签误分类的多状态马尔可夫模型场景（该场景可推广至隐马尔可夫模型（Hidden Markov Models, HMMs）），且贝叶斯计算无需借助计算成本高昂的数值梯度近似，即可获得极大似然估计（Maximum Likelihood Estimates, MLEs）。相关补充材料可在线获取。