Markov stick-breaking processes
本文提出一类新的随机离散分布构造方法——马尔可夫棍子断裂过程，通过引入马尔可夫依赖的断裂长度变量，扩展了经典的狄利克雷和皮特曼-约尔过程，并证明了其作为贝叶斯非参数模型的适用性。
Stick-breaking has a long history and is one of the most popular procedures for constructing random discrete distributions in statistics and machine learning. In particular, due to their intuitive construction and computational tractability they are ubiquitous in modern Bayesian nonparametric inference. Most widely used models, such as the Dirichlet and the Pitman–Yor processes, rely on i.i.d. or independent length variables. Here, we pursue a completely unexplored research direction by considering Markov length variables and investigate the corresponding general class of stick-breaking processes, which we term Markov stick-breaking processes. We establish conditions under which the associated species sampling process is proper and the distribution of a Markov stick-breaking process has full topological support, two fundamental desiderata for Bayesian nonparametric models. We also analyze the stochastic ordering of the weights and provide a new characterization of the Pitman–Yor process as the only stick-breaking process invariant under size-biased permutations, under mild conditions. Moreover, we identify two notable subclasses of Markov stick-breaking processes that enjoy appealing properties and include Dirichlet, Pitman–Yor and geometric priors as special cases. Our findings include distributional results enabling posterior inference algorithms and methodological insights.