On a class of completely random measures (CRMs) and its role in Bayesian analysis
schedule le lundi 27 avril 2020 de 17h00 à 18h00
Organisé par : F. Bechtold, W. Da Silva , A. Fermanian, S. Has, Y. Yu
Intervenant : Riccardo Passeggeri (LPSM)
Lieu : Online at : https://bigbluebutton.math.upmc.fr/b/ade-phf-9dg
Sujet : On a class of completely random measures (CRMs) and its role in Bayesian analysis
Quasi-infinitely divisible (QID) distributions have been recently introduced by Lindner, Pan and Sato (Trans. Amer. Math. Soc. 370 (2018) 8483-8520). A random variable X is QID if and only if there exist two infinitely divisible (ID) random variables Y and Z s.t. X +Y = Z (in distribution) and Y is independent of X.
In this talk, we present QID random measures, describe their properties and their role in Bayesian analysis.
In particular, we show that QID random measures are dense in the space of all CRMs with respect to convergence in distribution. In other words, any CRM can be approximated by a QID random measure. Moreover, we demonstrate that there exists a one to one correspondence between the law of QID random measures and certain characteristic pairs (as it happens for the Poisson point processes). That is, the characteristic function of a QID random measure has an explicit formulation written in terms of two (unique) deterministic objects. Finally, we present a nonparametric Bayesian statistical framework based on QID random measures, which extends recent works of Michael Jordan, Tamara Broderick, Trevor Campbell and co-authors.