Abstract

We define the beta diffusion tree, a random tree structure with a set of leaves that defines a colle tion of overlapping subsets of objects, known as a feature allocation. The generative process for the tree is defined in terms of particles (representing the objects) diffusing in some continuous space, analogously to the Dirichlet and Pitman–Yor diffusion trees (Neal, 2003b; Knowles & Ghahramani, 2011), both of which define tree structures over clusters of the particles. With the beta diffu sion tree, however, multiple copies of a particle may exist and diffuse to multiple locations in the continuous space, resulting in (a random number of) possibly overlapping clusters of the objects. We demonstrate how to build a hierarchicallyclustered 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 arrays, international development statistics, and intranational socioeconomic measurements.