arXiv subjects

Marta Catalano

Publications and source records attributed to Marta Catalano.

6 recordsLinked to original sources

arXiv2025

Measures of Dependence based on Wasserstein distances

Measuring dependence between random variables is a fundamental problem in Statistics, with applications across diverse fields. While classical measures such as Pearson's correlation have been widely used for over a century, they have notable limitations, particularly in capturing nonlinear relationships and extending to general metric spaces. In recent years, the theory of Optimal Transport and Wasserstein distances has provided new tools to define measures of dependence that generalize beyond Euclidean settings. This survey explores recent proposals, outlining two main approaches: one based on the distance between the joint distribution and the product of marginals, and another leveraging conditional distributions. We discuss key properties, including characterization of independence, normalization, invariances, robustness, sample, and computational complexity. Additionally, we propose an alternative perspective that measures deviation from maximal dependence rather than independence, leading to new insights and potential extensions. Our work highlights recent advances in the field and suggests directions for further research in the measurement of dependence using Optimal Transport.

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arXiv2025

Measuring Partial Exchangeability with Reproducing Kernel Hilbert Spaces

In Bayesian multilevel models, the data are structured in interconnected groups, and their posteriors borrow information from one another due to prior dependence between latent parameters. However, little is known about the behaviour of the dependence a posteriori. In this work, we develop a general framework for measuring partial exchangeability for parametric and nonparametric models, both a priori and a posteriori. We define an index that detects exchangeability for common models, is invariant by reparametrization, can be estimated through samples, and, crucially, is well-suited for posteriors. We achieve these properties through the use of Reproducing Kernel Hilbert Spaces, which map any random probability to a random object on a Hilbert space. This leads to many convenient properties and tractable expressions, especially a priori and under mixing. We apply our general framework to i) investigate the dependence a posteriori for the hierarchical Dirichlet process, retrieving a parametric convergence rate under very mild assumptions on the data; ii) eliciting the dependence structure of a parametric model for a principled comparison with a nonparametric alternative.

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arXiv2025

Hierarchical Random Measures without Tables

The hierarchical Dirichlet process is the cornerstone of Bayesian nonparametric multilevel models. Its generative model can be described through a set of latent variables, commonly referred to as tables within the popular restaurant franchise metaphor. The latent tables simplify the expression of the posterior and allow for the implementation of Gibbs sampling algorithms to approximately draw posterior samples. However, managing their assignments can become computationally expensive, especially as the size of the dataset and the number of levels increase. In this work, we identify a prior for the concentration parameter of the hierarchical Dirichlet process that (i) induces a quasi-conjugate posterior distribution, and (ii) removes the need for tables, leading to more interpretable expressions for the posterior, with both a scalable and an exact algorithm to sample from it. Remarkably, this construction extends beyond the Dirichlet process, leading to a new framework for defining normalized hierarchical random measures and a new class of algorithms to sample from their posteriors. The key analytical tool is the independence of multivariate increments, that is, their representation as completely random vectors.

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arXiv2024

Hierarchical Integral Probability Metrics: A distance on random probability measures with low sample complexity

Random probabilities are a key component to many nonparametric methods in Statistics and Machine Learning. To quantify comparisons between different laws of random probabilities several works are starting to use the elegant Wasserstein over Wasserstein distance. In this paper we prove that the infinite dimensionality of the space of probabilities drastically deteriorates its sample complexity, which is slower than any polynomial rate in the sample size. We propose a new distance that preserves many desirable properties of the former while achieving a parametric rate of convergence. In particular, our distance 1) metrizes weak convergence; 2) can be estimated numerically through samples with low complexity; 3) can be bounded analytically from above and below. The main ingredient are integral probability metrics, which lead to the name hierarchical IPM.

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arXiv2023

A Wasserstein index of dependence for random measures

Optimal transport and Wasserstein distances are flourishing in many scientific fields as a means for comparing and connecting random structures. Here we pioneer the use of an optimal transport distance between Lévy measures to solve a statistical problem. Dependent Bayesian nonparametric models provide flexible inference on distinct, yet related, groups of observations. Each component of a vector of random measures models a group of exchangeable observations, while their dependence regulates the borrowing of information across groups. We derive the first statistical index of dependence in $[0,1]$ for (completely) random measures that accounts for their whole infinite-dimensional distribution, which is assumed to be equal across different groups. This is accomplished by using the geometric properties of the Wasserstein distance to solve a max-min problem at the level of the underlying Lévy measures. The Wasserstein index of dependence sheds light on the models' deep structure and has desirable properties: (i) it is $0$ if and only if the random measures are independent; (ii) it is $1$ if and only if the random measures are completely dependent; (iii) it simultaneously quantifies the dependence of $d \ge 2$ random measures, avoiding the need for pairwise comparisons; (iv) it can be evaluated numerically. Moreover, the index allows for informed prior specifications and fair model comparisons for Bayesian nonparametric models.

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arXiv2023

Merging Rate of Opinions via Optimal Transport on Random Measures

Random measures provide flexible parameters for Bayesian nonparametric models. Given two different priors for a random measure, we develop a natural framework to investigate the rate at which the corresponding posteriors merge, as the sample size increases. We define a new distance between the laws of random measures that is built as a Wasserstein distance on the ground space of unbalanced measures, endowed with the bounded Lipschitz metric. We develop tight analytical bounds for its specification to completely random measures, including the special case of Poisson and gamma random measures. The bounds are interpreted in terms of an adapted extended Wasserstein distance between the L\'evy measures and are used to investigate the merging between the posteriors of normalized gamma and generalized gamma priors. After a careful study on the identifiability of the law of the random measure, interesting asymptotic and finite-sample insights are derived without putting any assumption on the true data generating process.

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