Bayesian Heterogeneous Hidden Markov Models with an Unknown Number of States

Hidden Markov models (HMMs) are valuable tools for analyzing longitudinal data due to their capability to describe dynamic heterogeneity. Conventional HMMs typically assume that the number of hidden states (i.e., the order of HMMs) is known or predetermined through criterion-based methods. However, prior knowledge about the order is often unavailable, and a pairwise comparison using criterion-based methods becomes increasingly tedious and computationally demanding when the model space enlarges. A few studies have considered simultaneously performing order selection and parameter estimation under the frequentist framework. Still, they focused only on homogeneous HMMs and thus cannot accommodate situations where potential covariates affect the between-state transition. This study proposes a Bayesian double-penalized (BDP) procedure to conduct a simultaneous order selection and parameter estimation for heterogeneous HMMs. We develop a novel Markov chain Monte Carlo algorithm coupled with an efficient adjust-bound reversible jump scheme to address the challenges in updating the order. Simulation studies show that the proposed BDP procedure considerably outperforms the commonly used criterion-based methods. An application to the Alzheimer’s Disease Neuroimaging Initiative study further confirms the utility of the proposed method. Supplementary materials for this article are available online.

隐马尔可夫模型（Hidden Markov Models，HMMs）凭借其可刻画动态异质性的能力，是分析纵向数据的极具价值的工具。传统隐马尔可夫模型通常假设隐状态的数量（即隐马尔可夫模型的阶数）已知，或是通过基于准则的方法预先确定。然而，关于模型阶数的先验知识往往难以获取，且当模型空间扩大时，采用基于准则的方法进行两两比较会愈发繁琐，同时计算成本也显著提升。已有少量研究考虑在频率学派框架下同步开展模型阶数选择与参数估计工作，但此类研究仅针对齐次隐马尔可夫模型，因此无法适配存在潜在协变量影响状态间转移的场景。本研究提出贝叶斯双惩罚（Bayesian Double-Penalized，BDP）方法，用于实现非齐次隐马尔可夫模型的阶数选择与参数估计同步进行。我们开发了一种全新的马尔可夫链蒙特卡洛（Markov Chain Monte Carlo，MCMC）算法，并结合高效的调整界可逆跳变机制，以解决模型阶数更新过程中的挑战。模拟实验结果表明，所提出的贝叶斯双惩罚方法的性能显著优于常用的基于准则的方法。通过对阿尔茨海默病神经影像倡议（Alzheimer’s Disease Neuroimaging Initiative，ADNI）研究数据的应用，进一步验证了所提方法的实用性。本文的补充材料可在线获取。