Beta diffusion trees and hierarchical feature allocations
This work addresses the need for flexible hierarchical clustering models that handle overlapping features, which is incremental as it builds upon existing diffusion tree methods.
The authors tackled the problem of modeling overlapping subsets of objects by introducing the beta diffusion tree, a random tree structure that generalizes the Dirichlet diffusion tree to allow objects to belong to multiple subsets, and demonstrated its application in hierarchically-clustered factor analysis with inference via Markov chain Monte Carlo, showing results on gene expression, international development, and socioeconomic datasets.
We define the beta diffusion tree, a random tree structure with a set of leaves that defines a collection of overlapping subsets of objects, known as a feature allocation. A generative process for the tree structure is defined in terms of particles (representing the objects) diffusing in some continuous space, analogously to the Dirichlet diffusion tree (Neal, 2003), which defines a tree structure over partitions (i.e., non-overlapping subsets) of the objects. Unlike in the Dirichlet diffusion tree, multiple copies of a particle may exist and diffuse along multiple branches in the beta diffusion tree, and an object may therefore belong to multiple subsets of particles. We demonstrate how to build a hierarchically-clustered factor analysis model with the beta diffusion tree and how to perform inference over the random tree structures with a Markov chain Monte Carlo algorithm. We conclude with several numerical experiments on missing data problems with data sets of gene expression microarrays, international development statistics, and intranational socioeconomic measurements.