Parametric Nonparametric Statistics: An Introduction to Mixtures of Finite Polya Trees

We present an introduction to an exciting approach to Bayesian nonparametrics, mixtures of Polya trees (MPTs). MPTs can be viewed as a simple generalization of standard parametric statistical distributions. MPTs use a partition of the support of the original distribution's density. The more general density retains the shape of the original distribution on each partition set but adds new parameters that are conditional probabilities. This provides a highly flexible family of distributions, one that is appropriate for nonparametric fitting. MPTs allow for data-driven features to emerge that are sometimes surprising, such as multimodality and skewness, and can vastly improve model fit relative to the original parametric family. Polya tree models are broadly applicable and easily programmed given existing MCMC schemes for fitting the original parametric model. Our examples include Paraguayan monkey hunting and toenail fungus treatment. In the first of these examples, we find that a normal theory model works quite well, but that there is little price to be paid for the extra generality of fitting a Mixture of Polya trees.