Nonparametric priors for finite unknown cardinalities of sampling spaces

The Dirichlet process and its popular generalization, the Pitman-Yor process, are often considered as priors in the context of multinomial sampling. They permit inferences on discrete sampling spaces of infinite cardinality. In fact, they a priori assume that there are infinitely many di erent outcomes to be observed, but rule out that there are only finitely many things. This, for instance, limits their usage in species sampling problems, where among other things we have to infer an unknown, but finite cardinality. Following first principles of inductive inference, such as exchangeability, we characterize a new class of nonparametric priors that extends the Pitman-Yor process, and permits the elicitation of a posterior distribution for the cardinality of the sample space, as well as the derivation of non-degenerate probabilities for any number of novel outcomes to appear given a finite sample. Over the course of the last decade, the Dirichlet process, and even more its generalization into the Pitman-Yor process, have met an indisputable success in the field of discrete Bayesian nonparametrics. One obvious explanation of this success lies in the analytical tractability of the inference procedures they support in settings where there is an infinite sample space to be considered. This benefit is certaintly best exemplified by the multiple uses of these processes as priors in clustering problems where the number of clusters is unknown, although applications where these processes drive the generation of the observations themselves have also gained a wide popularity. While meant to tackle very di erent problems, these applications share one common feature: only properties pertaining to finite samples are queried. In the context of clustering, one is indeed typically interested in the posterior distribution of the random partition of the observations into clusters; in the species sampling problem, the focus is rather on predicting the next outcome in a manner consistent with the possibility that an outcome unobserved so far might appear. Nonetheless, the inference of universal properties of the sample space, such as how many clusters would one encounter if one were to keep sampling indefinitely many times, or how many species are there in total, is usually avoided. This bias does not really follow from a lack of scientific interest in the answers to these questions, but rather from the full support provided by those processes to the universal hypothesis that infinitely many di erent outcomes would ultimately occur. In other words, any hypothesis set-