Asymptotic Inference for Exchangeable Gibbs Partitions

Abstract

We study the asymptotic properties of parameter estimation and predictive inference under the exchangeable Gibbs partition, characterized by a discount parameter $\alpha\in(0,1)$ and a triangular array $v_{n,k}$ satisfying a backward recursion. Assuming that $v_{n,k}$ admits a mixture representation over the Ewens--Pitman family $(\alpha, \theta)$, with $\theta$ integrated by an unknown mixing distribution, we show that the (quasi) maximum likelihood estimator $\hat\alpha_n$ (QMLE) for $\alpha$ is asymptotically mixed normal. This generalizes earlier results for the Ewens--Pitman model to a more general class. We further study the predictive task of estimating the probability simplex $\mathsf{p}_n$, which governs the allocation of the $(n+1)$-th item, conditional on the current partition of $[n]$. Based on the asymptotics of the QMLE $\hat{\alpha}_n$, we construct an estimator $\hat{\mathsf{p}}_n$ and derive the limit distributions of the $f$-divergence $\mathsf{D}_f(\hat{\mathsf{p}}_n||\mathsf{p}_n)$ for general convex functions $f$, including explicit results for the TV distance and KL divergence. These results lead to asymptotically valid confidence intervals for both parameter estimation and prediction.