The Bernstein-von Mises theorem for the Pitman-Yor process of nonnegative type

The Pitman-Yor process is a nonparametric species sampling prior with number of different species of the order of $n^\sigma$ for some $\sigma>0$. In case of an atomless true distribution, the asymptotic distribution of the posterior of the Pitman-Yor process was known but typically inconsistent. In this paper, we extend this result into a general theorem for arbitrary true distributions. For discrete distributions the posterior is consistent, but it turns out that there can be a bias which does not converge to zero at the $\sqrt{n}$ rate. We propose a bias correction and show that after correcting for the bias, the posterior distribution will be asymptotically normal. Without the bias correction, the coverage of the credible sets can be arbitrarily low, and we illustrate this finding with simulations where we compare the coverage of corrected and uncorrected credible sets.