Projected P\'olya Tree

One way of defining probability distributions for circular variables (directions in two dimensions) is to radially project probability distributions, originally defined on ${\mathbb{R}}^2$, to the unit circle. Projected distributions have proved to be useful in the study of circular and directional data. Although any bivariate distribution can be used to produce a projected circular model, these distributions are typically parametric. In this article we consider a bivariate P'olya tree on ${\mathbb{R}}^2$ and project it to the unit circle to define a new Bayesian nonparametric model for circular data. We study the properties of the proposed model, obtain its posterior characterisation and show its performance with simulated and real datasets.