Theory of Probability and Mathematical Statistics
Bayesian nonparametric inference on a Fréchet class
Emanuela Dreassi, Luca Pratelli and Pietro Rigo
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Abstract: Let (X,F,μ) and (Y,G,ν) be probability spaces and (Zn) be a sequence of random variables with values in (X×Y,F⊗G). Let Γ(μ,ν) be the collection of all probability measures p on F⊗G such that p(A×Y)=μ(A) and p(X×B)=ν(B) for all A∈F and B∈G.
In this paper, we build some probability measures Π on Γ(μ,ν). In addition, for each such Π, we assume that (Z
n) is exchangeable with de Finetti’s measure Π and we evaluate the conditional distribution Π(⋅|Z
1,...Z
n). In Bayesian nonparametrics, if (Z
1,...Z
n) are the available data, Π and Π(⋅|Z
1,...Z
n) can be regarded as the prior and the posterior, respectively. To support this interpretation, it suffices to think of a problem where the unknown probability distribution of some bivariate phenomenon is constrained to have marginals μ and ν. Finally, analogous results are obtained for the set Γ(μ) of those probability measures on F⊗G with marginalμ on F (but arbitrary marginal on G). That is, we introduce some priors on Γ(μ) and we evaluate the corresponding posteriors.
Keywords: Bayesian nonparametrics, copula, exchangeability, Fréchet class, mass transportation, random probability measure
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