Executive Vice President and Chief Statistician, NORCUniversity of ChicagoMay 2019-Present
Randomized Polya Trees: Bayesian Nonparametrics for Multivariate Data Analysis
The nonparametric approach to statistical modeling is appealing because it readily accommodates non-standard relationships in data. This dissertation is a first step to understanding the usefulness of Polya tree priors for modeling in multidimensional Euclidean spaces. In particular, the Polya tree prior is applied to a multidimensional Euclidean space. Using binary perpendicular recursive partitioning of a hypercube in $\Re^K$, it is shown that marginal distributions of Polya tree priors are Polya trees, and a conditional predictive distribution simulation scheme for exploring conditonal relationships among $K$ variables in a $K$-dimensional space is developed. Its usefulness for missing data imputation is also discussed. To address partition dependence -- a critical limitation of Polya trees -- the Randomized Polya tree is defined and developed. This new framework inherits the structure of Polya trees but induces smoothing of discontinuities in predictive distributions. Theoretical aspects of the new framework are developed, followed by discussion of methodological and computational issues arising in implementation. Analyses of two data sets highlight aspects of inference with randomized trees. Future directions for research are discussed.