Igor Prünster

Stick-breaking has a long history and is one of the most popular procedures for constructing random discrete distributions in statistics and machine learning. In particular, due to their intuitive construction and computational tractability they are ubiquitous in modern Bayesian nonparametric inference. Most widely used models, such as the Dirichlet and the Pitman–Yor processes, rely on i.i.d. or independent length variables. Here, we pursue a completely unexplored research direction by considering Markov length variables and investigate the corresponding general class of stick-breaking processes, which we term Markov stick-breaking processes. We establish conditions under which the associated species sampling process is proper and the distribution of a Markov stick-breaking process has full topological support, two fundamental desiderata for Bayesian nonparametric models. We also analyze the stochastic ordering of the weights and provide a new characterization of the Pitman–Yor process as the only stick-breaking process invariant under size-biased permutations, under mild conditions. Moreover, we identify two notable subclasses of Markov stick-breaking processes that enjoy appealing properties and include Dirichlet, Pitman–Yor and geometric priors as special cases. Our findings include distributional results enabling posterior inference algorithms and methodological insights.

We present a novel approach to ecological risk assessment by recasting the species sensitivity distribution (SSD) method within a Bayesian nonparametric (BNP) framework. Widely mandated by environmental regulatory bodies globally, SSD has faced criticism due to its historical reliance on parametric assumptions when modelling species variability. By adopting nonparametric mixture models, we address this limitation, establishing a statistically robust foundation for SSD. Our BNP approach offers several advantages, including its efficacy in handling small datasets or censored data, which are common in ecological risk assessment, and its ability to provide principled uncertainty quantification alongside simultaneous density estimation and clustering. We utilize a specific nonparametric prior as the mixing measure, chosen for its robust clustering properties, a crucial consideration given the lack of strong prior beliefs about the number of components. Through simulation studies and analysis of real datasets, we demonstrate the superiority of our BNP-SSD over classical SSD methods. We also provide a BNP-SSD Shiny application, making our methodology available to the Ecotoxicology community. Moreover, we exploit the inherent clustering structure of the mixture model to explore patterns in species sensitivity. Our findings underscore the effectiveness of the proposed approach in improving ecological risk assessment methodologies.

Stick-breaking has a long history and is one of the most popular procedures for constructing random discrete distributions in Statistics and Machine Learning. In particular, due to their intuitive construction and computational tractability they are ubiquitous in modern Bayesian nonparametric inference. Most widely used models, such as the Dirichlet and the Pitman-Yor processes, rely on iid or independent length variables. Here we pursue a completely unexplored research direction by considering Markov length variables and investigate the corresponding general class of stick-breaking processes, which we term Markov stick-breaking processes. We establish conditions under which the associated species sampling process is proper and the distribution of a Markov stick-breaking process has full topological support, two fundamental desiderata for Bayesian nonparametric models. We also analyze the stochastic ordering of the weights and provide a new characterization of the Pitman-Yor process as the only stick-breaking process invariant under size-biased permutations, under mild conditions. Moreover, we identify two notable subclasses of Markov stick-breaking processes that enjoy appealing properties and include Dirichlet, Pitman-Yor and Geometric priors as special cases. Our findings include distributional results enabling posterior inference algorithms and methodological insights.

Linear functionals, or means, of discrete random probability measures are a natural probabilistic object and the investigation of their properties have a long and rich history. They appear in several areas of mathematics, including statistics, combinatorics, special functions, excursions of stochastic processes and financial mathematics, among others. Most contributions have aimed at determining their distribution starting from a fully specified random probability. This work addresses the inverse problem: the identification of the base measure of a discrete random probability measure yielding a specific mean distribution. Available results concern only the Dirichlet case for specific choices of the concentration parameter. Here we address the problem in much greater generality and, besides considering new instances of Dirichlet processes, we cover the normalized stable process and the Pitman–Yor process. In addition to their theoretical interest, the results are of practical relevance to Bayesian nonparametric inference, where the law of a random probability measure acts as a prior distribution: often pre-experimental information is available about a finite-dimensional projection of the data generating distribution, such as the mean, rather than about an infinite-dimensional parameter. We further extend our findings to mixture models, ubiquitous in statistics and machine learning.

Filippo Ascolani, Antonio Lijoi, Igor Prünster; Ascolani, Lijoi and Prünster's contribution to the Discussion of “Root and community inference on the laten

Hierarchical models enjoy great popularity due to their ability to handle heterogeneous groups of observations by leveraging on their underlying common structure. In a Bayesian nonparametric framework, the hierarchy is introduced at the level of group-specific random measures, and then translated to the observations’ level via suitable transformations. In this work, we propose a new strategy to derive closed-form expressions for the marginal and posterior distributions of each group. Indeed, by directly inserting a suitable set of latent variables into the generative model for the data, we unravel a common core shared by the different hierarchical constructions proposed in the Bayesian nonparametric literature. Specifically, we identify a key identity that underlies these models and highlight its role in the derivation of quantities of interest.

Modelling of the dependence structure across heterogeneous data is crucial for Bayesian inference since it directly impacts the borrowing of information. Despite the extensive advances over the last two decades, most available proposals only allow for nonnegative correlations. We derive a new class of dependent nonparametric priors that can induce correlations of any sign, thus introducing a new and more flexible idea of borrowing of information. This is achieved thanks to a novel concept, which we term hyper-tie, and represents a direct and simple measure of dependence. We investigate prior and posterior distributional properties of the model and develop algorithms to perform posterior inference. Illustrative examples on simulated and real data show that our proposal outperforms alternatives in terms of prediction and clustering.