Bayesian Nonparametrics for Directional Statistics

TL;DR

This paper introduces a new Bayesian nonparametric approach for circular density estimation using trigonometric polynomial bases, demonstrating theoretical consistency and practical advantages over existing methods.

Contribution

It develops a novel circular density basis, constructs nonparametric priors, and establishes strong posterior consistency with adaptive convergence rates.

Findings

Bayesian estimators outperform existing circular density estimators in simulations.

The proposed framework ensures strong posterior consistency under weak assumptions.

The method exploits shape-preserving properties of the density basis.

Abstract

We introduce a density basis of the trigonometric polynomials that is suitable to mixture modelling. Statistical and geometric properties are derived, suggesting it as a circular analogue to the Bernstein polynomial densities. Nonparametric priors are constructed using this basis and a simulation study shows that the use of the resulting Bayes estimator may provide gains over comparable circular density estimators previously suggested in the literature. From a theoretical point of view, we propose a general prior specification framework for density estimation on compact metric space using sieve priors. This is tailored to density bases such as the one considered herein and may also be used to exploit their particular shape-preserving properties. Furthermore, strong posterior consistency is shown to hold under notably weak regularity assumptions and adaptative convergence rates are obtained in terms of the approximation properties of positive linear operators generating our models.