标题： 使用广义分层棒-breaking过程估计无限维转移概率矩阵 显示英文标题

标题： Estimation of An Infinite Dimensional Transition Probability Matrix Using a Generalized Hierarchical Stick-Breaking Process

摘要： 马尔可夫链为建模序列随机过程提供了基础框架，其中转移概率矩阵表征了状态演变的动力学特性。 虽然经典的估计方法，如最大似然和经验贝叶斯方法，在有限状态设置中是有效的，但在涉及可数无限或动态扩展状态空间的应用中变得不足，这些情况在自然语言处理、种群动力学和行为建模等领域经常出现。 在本工作中，我们引入了一种新颖的贝叶斯非参数框架，通过采用一类新的先验，称为广义分层杆Breaking先验，用于估计无限维转移概率矩阵。 这种先验扩展了传统的狄利克雷过程和杆Breaking构造，使得转移概率矩阵的建模具有高度灵活性。 所提出的方法在由稀疏性、高维度和未观察到的状态空间特征的设置中，提供了一种合理的推断转移概率的方法，从而促进了无限维转移概率矩阵的统计推断的发展。 显示英文摘要

摘要： Markov chains provide a foundational framework for modeling sequential stochastic processes, with the transition probability matrix characterizing the dynamics of state evolution. While classical estimation methods such as maximum likelihood and empirical Bayes approaches are effective in finite-state settings, they become inadequate in applications involving countably infinite or dynamically expanding state spaces, which frequently arise in domains such as natural language processing, population dynamics, and behavioral modeling. In this work, we introduce a novel Bayesian nonparametric framework for estimating infinite-dimensional transition probability matrices by employing a new class of priors, termed the Generalized Hierarchical Stick-Breaking prior. This prior extends traditional Dirichlet process and stick-breaking constructions, enabling highly flexible modelling of transition probability matrices. The proposed approach offers a principled methodology for inferring transition probabilities in settings characterized by sparsity, high dimensionality, and unobserved state spaces, thereby contributing to the advancement of statistical inference for infinite-dimensional transition probability matrices.