Pitman Yor Diffusion Trees for Bayesian Hierarchical Clustering.

David A. Knowles, Zoubin Ghahramani

Abstract: In this paper we introduce the Pitman Yor Diffusion Tree (PYDT), a Bayesian non-parametric prior over tree structures which generalises the Dirichlet Diffusion Tree <xref ref-type="bibr" rid="ref30" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[30]</xref> and removes the restriction to binary branching structure. The generative process is described and shown to result in an exchangeable distribution over data points. We prove some theoretical properties of the model including showing its construction as the continuum limit of a nested Chinese restaurant process model. We then present two alternative MCMC samplers which allow us to model uncertainty over tree structures, and a computationally efficient greedy Bayesian EM search algorithm. Both algorithms use message passing on the tree structure. The utility of the model and algorithms is demonstrated on synthetic and real world data, both continuous and binary.