SCHEDULE

All lectures are in room 11-13 in Casa Convalescencia. 

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*** The schedule may still slightly change ***

09:30-10:10 Benjamin Recht (UC Berkeley)

10:10-11:00 COFFEE BREAK

11:00-11:40 Samory Kpotufe (Columbia University)

Short break

12:00-12:40 Jianqing Fan (Princeton University)

*** FREE AFTERNOON

Wicher Bergsma (London School of Economics and Political Science)

Regression modelling with I-priors

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Standard objective priors, such as the Jeffreys prior, g-prior, or reference prior, have the drawback that they can only be used for finite dimensional parameters. We introduce a new objective prior based on the Fisher information, which can be used for both finite and infinite dimensional parameters. The posterior mean or mode then provides a regularized estimator of the parameter. The I-prior is generally defined as a maximum entropy prior and has the intuitively appealing property that the more information is available on a linear functional of the parameter, the larger the prior variance, and the smaller the influence of the prior mean on the posterior distribution.

The I-prior methodology is particularly attractive from a computational point of view for estimating a parametric or nonparametric regression function. The I-prior for a regression function is very simple, namely Gaussian with covariance kernel proportional to the Fisher information on the regression function. We use the I-prior methodology to give a unifying framework for estimating a variety of regression models, including varying coefficient, multilevel, longitudinal models, and models with functional covariates and responses. 

Advantages compared to competing methods, such as Gaussian process regression or Tikhonov regularization, are ease of estimation and model comparison. In particular, we develop an EM algorithm with a simple E and M step for estimating hyperparameters, facilitating estimation for complex models. We also propose a novel parsimonious model formulation, requiring a single scale parameter for each (possibly multidimensional) covariate and no further parameters for interaction effects. This simplifies estimation because fewer hyperparameters need to be estimated, and also simplifies model comparison of models with the same covariates but different interaction effects; in this case, the model with the highest estimated likelihood can be selected. 

Using a number of widely analysed real data sets we show that predictive performance of our methodology is competitive. An R-package implementing the methodology is available (Jamil, 2020).

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Steffen Lauritzen (University of Copenhagen)

Locally associated graphical models and mixed convex exponential families

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Abstract: The notion of multivariate total positivity has proved to be useful in finance and psychology but may be too restrictive in other applications. We propose a concept of local association, where highly connected components in a graphical model are positively associated and study its properties. Our main motivation comes from gene expression data, where graphical models have become a popular exploratory tool. The models are instances of what we term mixed convex exponential families and we show that a mixed dual likelihood estimator has simple exact properties for such families as well as asymptotic properties similar to the maximum likelihood estimator. We further relax the positivity assumption by penalizing negative partial correlations in what we term the positive graphical lasso. Finally, we develop a GOLAZO algorithm based on block-coordinate descent that applies to a number of optimization procedures that arise in the context of graphical models, including the estimation problems described above. We derive results on existence of the optimum for such problems. (based on the article arXiv:2008.04688)

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Andreas Maurer (Istituto Italiano di Technologia)

Useful bounds for U-statistics

Abstract: I will present a Bernstein inequality for nondegenerate U-statistics with bounded kernels. The inequality reduces to the standard Bernstein-inequality for order one, it gives the correct variance term and avoids the combinatorial catastrophy, which the decoupling techniques incur for U-statistics of higher order. The bound can be made data-dependent by estimation of the variance term, and it can be equally applied to computationally more attractive incomplete U-statistics. Time permitting I will also give a bound for the suprema of U-processes in terms of Gaussian width.

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