# Projected P\'olya Tree

## TL;DR

This paper introduces a Bayesian nonparametric model for circular data by projecting a bivariate Pólya tree distribution onto the unit circle, enabling flexible modeling of directional data.

## Contribution

It proposes a novel nonparametric circular distribution model based on projecting a bivariate Pólya tree, expanding Bayesian methods for directional data analysis.

## Key findings

- Model effectively captures circular data patterns
- Posterior characterization is analytically tractable
- Performs well on simulated and real datasets

## Abstract

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.