Markov Stick-breaking Processes

TL;DR

This work introduces Markov stick-breaking processes (MSBp) to extend Bayesian nonparametric priors by allowing Markov dependence among stick-breaking length variables. It establishes when MSBp produce proper, full-support random measures and derives posterior inference tools, showing that Pitman–Yor invariance under size-biased permutations remains the unique stable case under mild conditions. It then develops two tractable subclasses, Beta Markov stick-breaking processes (BMSBp) and Lazy Markov stick-breaking processes (LMSBp), which interpolate between Dirichlet, Pitman–Yor, Geometric, and CDSBp regimes and provide practical posterior algorithms. A third-party posterior study demonstrates improved density estimation and clustering performance of MSBp mixtures over standard DP or CDSBp models. The results offer a flexible, computable framework for Bayesian nonparametric mixtures with tunable dependence in the weights, broadening applicability to complex discrete distributions and mixture models.

Abstract

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 iid 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.

Figures (13)

• Figure 1: Illustration of Theorem \ref{['theo:MSB_sb_PY']}. The upper and lower solid lines stand for $L_{\varepsilon'}$ and $L_{\varepsilon^*}$, respectively. The red dashed line shows that $\bm{z}'$ is a reflection of $\bm{z}^*$ with respect to the identity function (dotted-dashed line).
• Figure 2: Tie probability $\tau_p$ as $\theta$ varies in $[0,10]$ for BMSBp for different values of $N$ where $v_j \sim {\mathsf{Be}}(1-\sigma,\theta+j\sigma)$. Moreover, $\sigma = 0.3$ in panel A and $\sigma = 0$ in panel B.
• Figure 3: Plots of the mappings $n \mapsto \mathbb{E}[K_n]$ and $n \mapsto {\mathsf{Var}}(K_n)$ for BMSBp with distinct values of $N$ where $v_j \sim {\mathsf{Be}}(1-\sigma,\theta+j\sigma)$ with $\sigma = 0.3,\theta = 2$ (A) and $\sigma = 0,\theta = 3$ (B) .
• Figure 4: Tie probability $\tau_p$ as $\theta$ varies in $[0,10]$ for LMSBp with distinct values of $\rho$ where $v_j \sim {\mathsf{Be}}(1-\sigma,\theta+j\sigma)$ with $\sigma = 0.3$ (A) and $\sigma = 0$ (B) .
• Figure 5: Displays of the mappings $n \mapsto \mathbb{E}[K_n]$ and $n \mapsto {\mathsf{Var}}(K_n)$ for LMSBp with distinct values of $\rho$ where $v_j \sim {\mathsf{Be}}(1-\sigma,\theta+j\sigma)$ with $\sigma = 0.3,\theta = 2$ (A) and $\sigma = 0,\theta = 3$ (B) .

Theorems & Definitions (90)

• Theorem 2.1
• Proposition 2.2
• Proposition 2.3
• Theorem 2.4
• Proposition 2.5
• Theorem 2.6
• Theorem 3.1
• Corollary 3.2
• Theorem 4.1
• Theorem 4.2