Hilbert-based Indices of Dependence Between Random Probability Measures for Distributional Data and Bayesian Nonparametrics

This thesis develops a general and practical framework for defining, estimating, and interpreting indices of dependence between random probability measures. The core idea is to embed each probability measure into a Hilbert space, enabling the application of classical multivariate dependence tools to the space of distributions. Building on this representation, we introduce a family of model-free indices of dependence that varies with the choice of the embedding and the choice of a summary of the induced cross-covariance operator. We focus on Wasserstein-based and kernel-based embeddings, characterizing the theoretical and numerical behavior of the indices at their extreme values. When the kernel mean embedding is composed with the trace of the cross-covariance operator, we obtain an index that generalizes the widely used set-wise correlation in Bayesian Nonparametrics, exhibiting several notable properties that do not hold for the standard set-wise correlation. Applications of our Hilbert-based indices include the hierarchical clustering of functional brain imaging, the study of the Hierarchical Dirichlet Process a posteriori, and the development of a principled model comparison between parametric and nonparametric Bayesian models.