SEMINARS IN STATISTICS @ COLLEGIO CARLO ALBERTO
Venerdì 04/04/2025, presso il Collegio Carlo Alberto, in Piazza Arbarello 8, Torino, si terrà il seguente seminario:
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12.00-13.00
Speaker: Sergios AGAPIOU (University of Cyprus)
Title: HEAVY-TAILED BAYESIAN NONPARAMETRIC ADAPTATION OVER BESOV SPACES
Abstract: We will consider the Bayesian recovery of an unknown function from direct observations polluted by white Gaussian noise, and we will be interested in studying the asymptotic performance of the posterior in the infinitely informative data limit, in terms of rates of contraction. We will be especially interested in priors which are adaptive to the smoothness of the unknown function. In the past decade, certain hierarchical and empirical Bayes procedures based on Gaussian process priors, have been shown to achieve adaptation to spatially homogenous smoothness. However, we have recently shown that Gaussian priors are suboptimal for spatially inhomogeneous unknowns, that is, functions which are smooth in some areas and rough or even discontinuous in other areas of their domain. Such unknowns are abundant in applications such as imaging, and can be modeled using Besov spaces, which generalize (the more widely known) Sobolev and Hölder spaces. In contrast to Gaussian priors, we have shown that (similar) hierarchical and empirical Bayes procedures based on Laplace (series) priors, achieve adaptation to both homogeneously and inhomogeneously smooth functions. All of these procedures involve the tuning of a hyperparameter of the Gaussian or Laplace prior. We will introduce Besov spaces and will recall their minimax theory developed in the mid-late 90’s and has various interesting features. After reviewing the above Bayesian results, we will present a new strategy for adaptation to smoothness based on heavy-tailed priors. Specifically, we will show that adaptive rates of contraction in the minimax sense (up to logarithmic factors) are achieved without tuning of any hyperparameters. This adaptation is achieved for both homogeneously and inhomogeneously smooth unknowns, in particular, we will show that the studied heavy-tailed priors are adaptive over all Besov spaces and for all L^p-losses, for p from 1 up to infinity. Extensive numerical simulations corroborating the theory will be presented as well. This is joint work with Masoumeh Dashti, Tapio Helin, Aimilia Savva and Sven Wang (Laplace priors), and Ismaël Castillo and Paul Egels (heavy-tailed priors)
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Sarà possibile il seminario anche in streaming: chiunque volesse collegarsi è pregato di inviare una email entro *mercoledì 02/04/2025* a matteo.giordano@unito.it .
Il webinar è organizzato dalla "de Castro" Statistics Initiative (www.carloalberto.org/stats) in collaborazione con il Collegio Carlo Alberto.
Cordiali saluti,