Mas 2012

Journées MAS 2012
Session “Bayesian Nonparametrics”

Organisateur : Dominique Bontemps (Université de Toulouse)

Description de la session :
C’est une session en anglais, plusieurs des intervenants étant étrangers.
La session présentera le point de vue fréquentiste sur les méthodes bayésiennes dans des cadres nonparamétriques et semiparamétriques. De nombreux articles récents s’inscrivent dans deux directions principales. La première concerne la concentration de l’a-posteriori, dans la lignée de [GGv00], avec un fort intérêt pour les processus gaussiens ; la seconde, dans la suite de [Gho99], porte sur la normalité asymptotique de l’a-posteriori.
Enfin, le dernier exposé s’intéresse à des tests de monotonicité construits à partir de régions de crédibilité bayésiennes.

 

Les exposés :

  1. Judith Rousseau, Université Paris Dauphine (intervenante principale)

    Titre : On some aspects of frequentist properties of Bayesian non and semi-parametric approaches

  2. Catia Scricciolo, Université Bocconi, Milan

    Titre : Bayes and empirical Bayes: do they merge?

    Résumé : Bayesian inference is attractive for its coherence and good frequentist properties. However, it is a common experience that eliciting a honest prior may be difficult and, in practice, people often take an empirical Bayes approach, plugging empirical estimates of the prior hyperparameters into the posterior distribution. Even if not rigorously justified, the underlying idea is that, when the sample size is large, empirical Bayes leads to “similar” inferential answers. Yet, precise mathematical results seem to be missing. In this work, we give a more rigorous justification in terms of merging of Bayes and empirical Bayes posterior distributions. We consider two notions of merging: Bayesian weak merging and frequentist merging in total variation. Since weak merging is related to consistency, we provide sufficient conditions for consistency of empirical Bayes posteriors. Also, we show that, under regularity conditions, the empirical Bayes procedure asymptotically selects the value of the hyperparameter for which the prior mostly favors the “truth”. Examples include empirical Bayes density estimation with Dirichlet process mixtures.

    (Joint work with Sonia Petrone and Judith Rousseau)

  3. Bas Kleijn, Université d’Amsterdam

    Titre : Semiparametric posterior limits for regular and some irregular problems

    Résumé : We consider the Bayesian procedure from the frequentist perspective with a focus on marginal posterior limit distributions in regular (LAN) and some irregular (LAE) semiparametric estimation problems. In the early 1950’s Le Cam established the celebrated Bernstein-Von Mises theorem for regular (LAN) parametric estimation problems. In this talk, we extend Le Cam’s theorem to regular semiparametric context, showing that the marginal posterior for the parameter of interest converges to the sampling distribution of any efficient semiparametric point-estimator, under straightforward conditions on model and prior. The methodology can be extended further to a class of irregular semiparametric estimation problems introduced by Ibrahimov and Hasminski which are not smooth but display what is called locally asymptotically exponentiality (LAE). We establish an exponential Bernstein-Von Mises limit for the marginal posterior in LAE semiparametric problems.

    (Based on collaborations with P. Bickel and B. Knapik)

  4. J.-B. Salomond, Université Paris Dauphine (jean.bernard.salomond@ensae.fr)

    Titre : Adaptive Bayes test for monotonicity

    Résumé : We study the asymptotic behaviour of a Bayesian nonparametric test of qualitative hypotheses. More precisely, we focus on the problem of testing the monotonicity of a regression function, following an on going work of [MR11]. Even if some results are known in the frequentist framework, no Bayesian testing procedure has been proposed, at least none has been studied theoretically. The test proposed in this work is straightforward to implement, which is a great advantage compared to the frequentist tests proposed in the literature. We prove consistency of our testing procedure and also derive an upper bound on its separation rate. The separation rate is the rate at which the minimal distance between the null and a given alternative, for which the test procedure is consistent, decreases with the number of observations. In our case, these alternative classes of regression functions will be defined in terms of Hölder smoothness and in terms of their distance to the set of monotone regression functions. We also prove that our procedure is adaptive, in the sense that it does not depend on the smoothness of the regression function under the alternative, and leads to the optimal (up to a log factor) separation rate, for all classes of Hölder function with smoothness bounded by 1.

References

[GGv00]   Subhashis Ghosal, Jayanta K. Ghosh, and Aad W. van der Vaart. Convergence rates of posterior distributions. Ann. Statist., 28(2):500–531, 2000.

[Gho99]    Subhashis Ghosal. Asymptotic normality of posterior distributions in high-dimensional linear models. Bernoulli, 5(2):315–331, 1999.

[MR11]    R. McVinish and J. Rousseau. Bayesian testing of decreasing densities. 2011.