*Recent advances in Bayesian Nonparametrics*

Organizer: **Igor Pruenster**, Bocconi University

Bayesian Nonparametrics is currently one of the most active research streams in Statistics. During the last decade it has experienced an amazing growth in terms of both research output and size of the community. As for the latter, the field now comprises also leading experts in Frequentist Statistics and Probability Theory on the theoretical side and Biostatisticians, Computer Scientists and Engineers on the applied side. This makes Bayesian Nonparametrics a lively and exciting research field. In this session some of the recent developments concerning theory, modeling and computation are showcased.

**Posterior contraction for mixtures of product distributions****Long Nguyen,** University of Michigan, USA

Mixtures of product distributions arise as the model for sequences of exchangeable observations, a consequence of de Finetti's theorem. They are also a common choice for modeling heterogeneous groups of data populations. In this work we study the posterior contraction behavior of the mixing measure as the number of exchangeable sequences increases. Of particular interest is the role of the length of the exchangeable sequences, which may or may not increase, where the posterior contraction behavior will be tightly dependent on. We shall present results for mixture models covering a large class of density kernels for both continuous and discrete valued data.

This work is joint with Yun Wei.

*epsilon-Approximation to the Pitman-Yor process***Pierpaolo De Blasi,** Univerisity of Torino, Italy

We consider approximations to the popular Pitman-Yor process obtained by truncating the stick-breaking representation. The truncation is determined by a random stopping rule that achieves an almost sure control on the approximation error in total variation distance. We derive the asymptotic distribution of the random truncation point as the approximation error epsilon goes to zero in terms of a polynomially tilted positive stable random variable. The practical usefulness and effectiveness of this theoretical result is demonstrated by devising a sampling algorithm to approximate functionals of the epsilon-version of the Pitman-Yor process.

*Bayesian inferences on uncertain ranks and orderings***Andrés Felipe Barrientos,** Duke University, USA

It is common to be interested in rankings or order relationships among entities. In complex settings where one does not directly measure a univariate statistic upon which to base ranks, such inferences typically rely on statistical models having entity-specific parameters. The current literature struggles to present summaries of order relationships which appropriately account for uncertainty. A single estimated ranking can be highly misleading, particularly as it is common that the entities do not vary widely in the trait being measured, leading to large uncertainty and instability in ranking a moderate to large number of them. We observed such problems in attempting to rank player abilities based on data from the National Basketball Association (NBA). Motivated by this, we propose a general strategy for characterizing uncertainty in inferences on order relationships among parameters. Our approach adapts to scenarios in which uncertainty in ordering is high by producing more conservative results that improve interpretability. This is achieved through a reward function within a decision-theoretic framework. We show that our method is theoretically sound and illustrate its utility using simulations and an application to NBA player ability data.

*Measures of Dependence in Bayesian Nonparametrics***Marta Catalano,** Bocconi University, Italy

Completely random measures are the basic building block of the large majority of discrete priors for Bayesian nonparametric inference. More recently, there has been a growing interest in dependent nonparametric priors for modeling heterogeneous data: these are naturally obtained by inducing dependence at the level of the underlying completely random measures. Despite being widely used, the analysis of their dependence structure is usually confined to measures of linear correlation. By relying on the Wasserstein distance, we here propose to measure the dependence of a vector of completely random measures as the distance from the maximally dependent one in the same Fréchet class. A compound Poisson approximation is used to achieve bounds for the Wasserstein distance in terms of the underlying Lévy measure. These are then specialized to noteworthy examples in the Bayesian literature, where vectors of random measures represent a common tool to induce dependence among nonparametric priors for partially exchangeable data.