Measuring dependence under partial exchangeability with reproducing kernel Hilbert spaces
Partial exchangeability, first introduced by De Finetti, is a natural assumption in Bayesian Nonparametrics when the observations are grouped in a finite number of blocks with similar features. The distribution of each block is modeled with a random probability measure, leading to a vector of joint de Finetti measures. Given two groups, the dependence structure can show a continuous spectrum of behaviors between two extremes: either total independence or almost sure equality of the two random probability measures. We propose a correlation index based on reproducing kernel Hilbert spaces that measures how far a partially exchangeable model is to the two extreme cases under general assumptions. Moreover, a) it generalizes the set-wise linear correlation, which is the most common measure used in the Bayesian nonparametric literature for this scope; b) it can be applied to both prior and posterior distributions; c) it preserves analytical tractability under mixing. These results are then specialized for various popular Bayesian nonparametric models, such as the Hierarchical Dirichlet Process and Dependent Dirichlet Processes. Numerical simulations confirm our theoretical understanding of our proposal as a measure of dependence.
Area: CS15 - Recent Advances in Bayesian Nonparametric Statistics (Marta Catalano and Beatrice Franzolini)
Keywords: Partial Exchangeability, Dependence Index, Borrowing of Information, Hierarchical Dirichlet Process, Dependent Dirichlet Process, Reproducing Kernel Hilbert Spaces
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