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Random vectors of measures are the main building block to a major portion of Bayesian nonparametric models. The introduction of infinite–dimensional parameter spaces guarantees notable flexibility and generality to the models but makes their treatment and interpretation more demanding. To overcome these issues we seek a deep understanding of infinite–dimensional random objects and their role in modeling complex dependence structures in the data. Comparisons with baseline models play a major role in the learning process and are expressed through the introduction of suitable distances. In particular, we define a distance between the laws of random vectors of measures that builds on the Wasserstein distance and combines intuitive geometric properties with analytical tractability. This is first used to evaluate approximation errors in posterior sampling schemes and then culminates in the definition of a new principled and non model–specific measure of dependence for partial exchangeability, going beyond current measures of linear dependence. The study of dependence is complemented by the investigation of asymptotic properties for partially exchangeable mixture models from a frequentist perspective. We extend Schwartz theory to a multisample framework by relying on natural distances between vectors of densities and leverage it to find optimal contraction rates for a wide class of hierarchical models.

On Complex Dependence Structures in Bayesian Nonparametrics: a Distance–based Approach



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Settore SECS-S/01 - Statistica
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Descrizione: Marta_Catalano_Thesis.pdf
Tipologia: Tesi di dottorato
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