Dependent Stick-breaking processes and related random probability measures

Music City Center 

Stick-breaking has a long history and represents the most popular procedure 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 iid or independent length variables. Here we pursue a completely unexplored research direction by considering exchangeable and Markov length variables and investigate the corresponding general class of stick-breaking processes. We derive conditions ensuring that the associated species sampling process is proper and 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 notable subclasses of the new stick-breaking processes, that enjoy appealing properties and include Dirichlet, Pitman-Yor and Geometric priors as special cases. Our findings further include distributional results that allow us to develop computational algorithms for posterior inference. Interesting methodological implications are drawn from numerical implementations of Markov stick-breaking processes.