B1483
Title: A novel approach to (strong) posterior contraction rates via Wasserstein dynamics
Authors: Emanuele Dolera - University of Pavia (Italy) [presenting]
Abstract: Posterior contractions rates (PCRs) for Bayesian consistency are considered. The statistical model is a family $M = {f_t}_{t in T}$ of densities with $T$ included in a separable Hilbert space, possibly infinite-dimensional. Two main approaches have been developed: the former considers neighborhoods of the true density $f_{t_0}$ in the space of densities; the latter neighborhoods of the true value of the parameter $t_0$ in $T$. We follow the latter when the Hilbertian metric on $T$ is stronger than the one induced by restricting to $M$ usual metrics on densities ($L_p$, Hellinger, Kullback-Leibler, chi-square). We present two main statements, valid when: 1) we dispose of a Banach space-valued sufficient statistics which is classically consistent; 2) only the empirical distribution is at disposal as sufficient statistics. Critical to our approach is an assumption of Lipschitz-continuity for the posterior with respect to observed data, ensuing from the dynamic formulation of the Wasserstein distance. This sets forth a connection between PCRs and other problems: Laplace methods for integrals, Sanovs principle, rates of mean Glivenko-Cantelli theorems, and estimates of weighted Poincar-Wirtinger constants. As a complement, we present novel improvements in evaluating both Laplace integrals and Poincar-Wirtinger constants in infinite dimensions. Finally, we illustrate our method by explicitly evaluating the PCRs for logistic-Gaussian model and linear regression.
