Title: Bayesian Nonparametric Models and Its Applications

Abstract: Polya tree priors are random probability measures that are easily centered at standard parametric families, such as the normal. As such, they provide a convenient avenue toward creating a parametric/nonparametric model for data. Briefly, we center a Polya tree at an initial parametric guess on data; then by adding more details via data, departures from the initial guess will be captured and used for adjusting the guess to obtain a robust nonparametric estimation. We utilize the Polya tree for the classic problem of testing whether data distributions are the same across several subpopulations.  Test-statistics that are (empirical) Bayes factors constructed from independent Polya tree priors are proposed. Generalizations to censored and multivariate data are provided. The conceptually simple test statistic fares surprisingly well against competitors in simulations.

In addition, we propose a novel weighted multivariate Polya tree and use it in control scheme for detecting small mean and/or variance shifts in various types of control processes. By further weighting the Polya tree in the test statistic, the control chart now can heighten the sensitivity of detecting one or more out of control characteristics. Examples show that our chart performs well in monitoring a multivariate process where the normality assumption is violated.