This paper deals with suitable quantifications in approximating a probability measure by an “empirical” random probability measure p_n, depending on the first n terms of a sequence of random elements. Section 2 studies the range of oscillation near zero of the Wasserstein distance between p_0 and such an empirical measure, assuming the original random elements are i.i.d. from p_0. In Theorem 2.1 p_0 can be fixed in the space of all probability measures on R^d and p_n coincides with the empirical measure. In Theorem 2.2 (Theorem 2.3, respectively), p_0 is a d-dimensional Gaussian distribution (an element of a distinguished statistical exponential family, respectively) and p_n is another d-dimensional Gaussian distribution with estimated mean and covariance matrix (another element of the same family with an estimated parameter, respectively). These new results improve on allied recent works by providing also uniform bounds with respect to n. In Section 3, assuming the random elements are exchangeable, one studies the range of oscillation near zero of the Wasserstein distance between the conditional distribution – also called posterior – of the directing measure of the sequence, given the first n observations, and the point mass at pˆn. Similarly, an upper bound for the approximation of predictive distributions is given. Finally, Theorems from 3.3 to 3.5 reconsider Theorems from 2.1 to 2.3, respectively, according to a Bayesian perspective.

Uniform rates of the Glivenko-Cantelli convergence and their use in approximating Bayesian inferences.

Dolera Emanuele;Regazzini Eugenio
2019-01-01

Abstract

This paper deals with suitable quantifications in approximating a probability measure by an “empirical” random probability measure p_n, depending on the first n terms of a sequence of random elements. Section 2 studies the range of oscillation near zero of the Wasserstein distance between p_0 and such an empirical measure, assuming the original random elements are i.i.d. from p_0. In Theorem 2.1 p_0 can be fixed in the space of all probability measures on R^d and p_n coincides with the empirical measure. In Theorem 2.2 (Theorem 2.3, respectively), p_0 is a d-dimensional Gaussian distribution (an element of a distinguished statistical exponential family, respectively) and p_n is another d-dimensional Gaussian distribution with estimated mean and covariance matrix (another element of the same family with an estimated parameter, respectively). These new results improve on allied recent works by providing also uniform bounds with respect to n. In Section 3, assuming the random elements are exchangeable, one studies the range of oscillation near zero of the Wasserstein distance between the conditional distribution – also called posterior – of the directing measure of the sequence, given the first n observations, and the point mass at pˆn. Similarly, an upper bound for the approximation of predictive distributions is given. Finally, Theorems from 3.3 to 3.5 reconsider Theorems from 2.1 to 2.3, respectively, according to a Bayesian perspective.
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11571/1287370
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 7
  • ???jsp.display-item.citation.isi??? 5
social impact