In this talk we will consider inference with high dimensional data. We propose new methods for estimating and constructing confidence regions for a regression parameter of primary interest alpha_0, a parameter in front of the regressor of interest, such as the treatment variable or a policy variable. We show how to apply these methods to Z-estimators (for example, logistic regression and quantile regression). These methods allow to estimate alpha_0 at the root-n rate when the total number p of other…

Achieving balance between experimental groups is a cornerstone of causal inference. Without balance any observed difference may be attributed to a difference other than the treatment alone. In controlled/clinical trials, where the experimenter controls the administration of treatment, complete randomization of subjects has been the golden standard for achieving this balance because it allows for unbiased and consistent estimation and inference in the absence of any a priori knowledge or measurements. However, since estimator variance under complete randomization may be…

Superposition codes are asymptotically capacity achieving scheme for the Additive White Gaussian Noise channel. I will first show how a practical iterative decoder can be built based on a Belief Propagation type approach, closely related to the one performed in compressed sensing and sparse estimation problems. Secondly, I will show how the idea of spatial coupling in this context allows to built efficient and practical capacity achieving coding and decoding schemes. The links between the present problem, sparse estimations, and…

A central problem in analyzing networks is partitioning them into modules or communities, clusters with a statistically homogeneous pattern of links to each other or to the rest of the network. A principled approach to address this problem is to fit the network on a stochastic block model, this task is, however, intractable exactly. In this talk we discuss application of belief propagation algorithm to module detection. In the first part we present an asymptotically exact analysis of the stochastic…

While much attention has been paid recently to the construction of optimal algorithms that adaptively estimate low-dimensional parameters (described by sparsity, low-rank, or smoothness) in high-dimensional models, the theory of statistical inference and uncertainty quantification (in particular hypothesis tests & confidence sets) is much less well-developed. We will discuss some perhaps surprising impossibility results in the basic high-dimensional compressed sensing model, and some of the recently remerging positive results in the area.

We will discuss recent moment bounds and concentration inequalities for sample covariance operators based on a sample of n i.i.d. Gaussian random variables taking values in an infinite dimensional space. These bounds show that the size of the operator norm of the deviation of sample covariance from the true covariance can be completely characterized by two parameters: the operator norm of the true covariance and its so called "effective rank". These results rely on Talagrand's generic chaining bounds and on…

Many nonlinear Econometric models show evidence of weak identification. In this paper we consider minimum distance statistics and show that in a broad class of models the problem of testing under weak identification is closely related to the problem of testing a curved null in a finite-sample Gaussian model. Using the curvature of the model, we develop new finite-sample bounds on the distribution of minimum-distance statistics, which we show can be used to detect weak identification and to construct tests…

Large covariance matrix estimation is crucial for high-dimensional statistical inferences, and has also played an central role in factor analysis. Applications are found in analyzing financial risks, climate data, genomic data and PCA, etc. Commonly used approaches to estimating large covariances include shrinkages and sparse modeling. This talk will present new theoretical results on estimating large (inverse) covariance matrices under large N large T asymptotics, with a focus on the roles it plays in statistical inferences for large panel data…

The celebrated Berry-Esseen theorem, and its variants, provide a useful approximation to the sum of independent random variables by a Gaussian. In this talk, I will restrict attention to the important case of sums of integer random variables, arguing that Berry-Esseen theorems fall short from characterizing their general structure. I will offer stronger finitary central limit theorems, tightly characterizing the structure of these distributions, and show their implications to learning. In particular, I will present algorithms that can learn sums…

We explore the link between the Monge-Kantorovich problem and the Skorohod embedding problem. This question arises in particular in Mathematical Finance when seeking model-free bounds on some option prices when the marginal distributions of the underlying at various maturities are implied by European options prices. We provide a stochastic control approach which we connect to several important constructions. Finally we revisit in this light the celebrated Azéma-Yor solution of the Skorohod embedding problem. This talk is based on joint works…