超越时间齐次性：连续时间多状态马尔可夫模型
Beyond time-homogeneity for continuous-time multistate Markov models
AI总结：
本文针对连续时间多状态马尔可夫模型，指出分段时间齐次假设可能导致参数估计偏差，提出真正时间非齐次的似然计算方法，并适用于隐马尔可夫模型，通过贝叶斯计算避免数值梯度近似。
摘要：
多状态马尔可夫模型是对有限状态空间上观测或潜在随机过程进行数据建模的一种经典参数化方法。连续时间马尔可夫过程描述的是随时间不规则观测的数据，例如纵向医学数据中常见的情况。假设连续时间马尔可夫过程是时间齐次的，则可以从科尔莫戈罗夫前向方程（一个具有众所周知的矩阵指数解的微分方程组）推导出闭式似然函数。然而，不幸的是，前向方程对于连续时间、时间非齐次的马尔可夫过程没有解析解，因此研究人员和从业者常常做出简化假设，即过程是分段时间齐次的。在本文中，我们提供了在更现实的情景下（即分段齐次假设被违反时）参数估计可能产生的潜在偏差的直观理解和示例，并倡导一种真正时间非齐次方式的似然计算方法。我们特别关注允许状态标签错误分类的多状态马尔可夫模型，这更广泛地适用于隐马尔可夫模型（HMMs），并且贝叶斯计算避免了为获得最大似然估计（MLEs）而进行计算密集型的数值梯度近似的必要性。补充材料可在网上获取。
英文摘要：
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-inhomogeneous 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.
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