Pitman Yor Diffusion Trees for Bayesian Hierarchical Clustering.

Area of Science:

• Computational statistics
• Bayesian non-parametrics
• Machine learning

Background:

• Existing Bayesian non-parametric priors for tree structures, such as the Dirichlet Diffusion Tree, are limited to binary branching.
• There is a need for more flexible tree structure models that can handle non-binary branching and complex data.

Purpose of the Study:

• To introduce the Pitman Yor Diffusion Tree (PYDT), a novel Bayesian non-parametric prior over tree structures.
• To generalize existing diffusion tree models by removing the binary branching restriction.
• To provide efficient algorithms for modeling uncertainty in tree structures.

Main Methods:

• The generative process of the PYDT is described, demonstrating its resulting exchangeable distribution over data points.
• Theoretical properties are proven, including its construction as a continuum limit of a nested Chinese restaurant process.
• Two MCMC samplers and a greedy Bayesian EM search algorithm are presented, utilizing message passing on the tree structure.

Main Results:

• The PYDT model is shown to be a flexible generalization of existing tree priors.
• The proposed MCMC samplers and EM algorithm enable efficient modeling of uncertainty over tree structures.
• The utility of the PYDT and its algorithms is demonstrated on both synthetic and real-world continuous and binary data.

Conclusions:

• The Pitman Yor Diffusion Tree (PYDT) offers a powerful and flexible Bayesian non-parametric prior for modeling tree structures.
• The developed algorithms provide efficient methods for inference and uncertainty quantification in PYDT models.
• The PYDT demonstrates broad applicability across various data types and complex modeling tasks.