Stein's Method for Probability Distributions on $\mathbb{S}^1$

In this paper, we propose a modification to the density approach to Stein's method for intervals for the unit circle $\mathbb{S}^1$ which is motivated by the differing geometry of $\mathbb{S}^1$ to Euclidean space. We provide an upper bound to the Wasserstein metric for circular distributions and exhibit a variety of different bounds between distributions; particularly, between the von-Mises and wrapped normal distributions, and the wrapped normal and wrapped Cauchy distributions.