Smoothing of Bayesian forest estimators in density estimation

schedule le lundi 08 juin 2020 de 17h00 à 18h00

Organisé par : F. Bechtold, W. Da Silva , A. Fermanian, S. Has, Y. Yu

Intervenant : Thibault Randrianarisoa (LPSM)
Lieu : Online at https://bigbluebutton.math.upmc.fr/b/ade-phf-9dg

Sujet : Smoothing of Bayesian forest estimators in density estimation

Résumé :

With CART regression trees, the values of the estimator in the leaf cells are equal to the mean of the observed data in each cell. Hence, once the tree structure is specified, this defines a histogram estimator. As a drawback, this limits its performance when the regression function f0 to be recovered is α- Hölder regular, with α > 1. However, it has been noted that ensemble methods, defining a forest estimator, could generate a smoothing effect, improving over the convergence rate of single tree estimators.
We are going to see how such ideas translate into nonparametric Bayesian inference. Indeed, a popular prior distribution involving a tree structure and used in density estimation is the Ploya tree prior. So we will see how we can define prior distributions via forests of Polya trees. First, these lead to nearly optimal contraction rates of the posterior distribution, for any regularity α > 0 of the true density f0. Secondly, such priors can also be made adaptive to α > 0. This is a sought-after property of inference methods as this smoothness parameter is often unknown in practice.