Beyond Time-Homogeneity for Continuous-Time Multistate Markov Models

<i>Abstract–</i>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 article, 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.

摘要——多状态马尔可夫模型（Multistate Markov models）是针对定义于有限状态空间的观测或潜随机过程的数据建模的经典参数化方法。连续时间马尔可夫过程可用于描述以非规则时间间隔进行观测的数据，这在纵向医疗数据中尤为典型。若假设连续时间马尔可夫过程为时间齐次的，则可通过柯尔莫哥洛夫前向方程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。但遗憾的是，对于连续时间、时间非齐次的马尔可夫过程，前向方程并无解析解，因此研究者与实务人员通常会做出简化假设，即该过程为分段时间齐次的。本文阐述了在违反分段齐次性假设的更具现实性的场景中，参数估计可能产生的潜在偏倚，并提出了一种真正采用时间非齐次模式的似然计算方案。本文特别聚焦于允许存在状态标签误分类的多状态马尔可夫模型场景，该场景可更广泛地推广至隐马尔可夫模型（HMMs）；同时，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取最大似然估计（MLEs）。本文补充材料可在线获取。