Schedule for Fall 2021

9/22/21

Summer Intern Workshop

Yuqi Gu (Columbia Stats)

Title: “Deep, Graphical, and Identifiable Modeling of Latent Structures and Their Application”

Abstract: This talk covers some of my recent work on latent variable models, including in particular the Bayesian Pyramids: identifiable deep discrete latent structure models for discrete data. Multivariate discrete data are routinely collected in biomedical and social sciences. It is of great importance to build interpretable parsimonious models that perform dimension reduction and uncover meaningful latent structures from such discrete data. Identifiability is a fundamental requirement for valid modeling and inference, yet is challenging to address when there are complex latent structures. We propose a class of identifiable deep discrete latent structure models termed Bayesian pyramids. Theoretically, we establish the identifiability of Bayesian pyramids by developing transparent conditions on the pyramid-shaped multilayer latent graph. Methodologically, we focus on the two-latent-layer model and propose a Bayesian shrinkage estimation approach. Simulation results corroborate the identifiability and estimability of the model parameters. Application of the methodology to DNA nucleotide sequence data uncovers useful discrete latent features that are highly predictive of sequence types. The proposed framework provides a recipe for interpretable unsupervised learning of discrete data, and can be a useful alternative to popular machine learning methods.

Along the way, I will also talk about the interesting connections between my research and tensor decompositions, mixed membership models, and deep generative models.

Attention: The Student Seminar will be in hybrid mode this semester. Most talks and events will be held in person, and people can also join via zoom. In-person participation is only available to Columbia affiliates with building access.

Join URL: https://columbiauniversity.zoom.us/j/91860272586?pwd=OWFyTjZNQko0Z3ZzZ2Y5bFZKYUNaQT09

Meeting ID: 918 6027 2586

Passcode: 225317        

10/6/21

Yi-Hsuan Lee (Educational Testing Service)

Title: Application of Statistics in Educational Measurement

Abstract: Educational measurement is a field that uses educational tests to obtain observations from
test takers in order to make inferences about their knowledge and skills. Measurement theory has been
developed to provide the foundation for evaluating educational tests and their uses and interpretations,
including fundamental concepts, considerations, and various statistical models and techniques to
quantify those concepts and considerations. In this presentation, I will provide a brief introduction of
the key concepts and some considerations in measurement theory. I will also discuss how statistical
methods may be applied to assess the quality of educational tests and to infer the knowledge and skills
of interest to educational researchers and practitioners, with some focus on more recent challenges and
opportunities in the field.

Morgane Austern (Harvard)

Title: Some recent results for model evaluation

Abstract: Estimating and evaluating the generalization capabilities of an estimator is a fundamental task of statistical inference. In this talk we are interested in better understanding how well the cross-validated risk estimates the risk, and in improving finite sample generalization bounds.

In the first part of this talk, we study the cross-validation method, an ubiquitous method for risk estimation, and establish its asymptotic properties for a large class of models and with an arbitrary number of folds. Under stability conditions, we establish a central limit theorem and Berry-Esseen bounds for the cross-validated risk, which enable us to compute asymptotically accurate confidence intervals. Using our results, we study the statistical speed-up offered by cross validation compared to a train-test split procedure. We reveal some surprising behavior of the cross-validated risk and establish the statistically optimal choice for the number of folds. In the second part of this talk, we remark that while concentration inequalities are fundamental tools to obtain finite sample generalization guarantees, those bounds are known to be loose. Moreover, while limit theorems provide asymptotically tight bounds, those tools are not valid for finite samples. Motivated by this observation, we propose a new method for deriving concentration inequalities that is both valid in finite samples and asymptotically optimal. We demonstrate that the bounds obtained improve on classical concentration inequalities such as the Bernstein or Azuma inequality.

11/17/21

Hongseok Namkoong (Columbia Business School, DRO)

Nabarun Deb (Columbia Stats)

Title: Optimal transport in statistics and Pitman efficient multivariate distribution-free testing

Abstract: In recent years, the problem of optimal transport (see e.g., Villani (2003)) has received significant attention in statistics and machine learning due to its powerful geometric properties. In this talk, we introduce the optimal transport problem and present concrete applications of this theory in statistics. In particular, we will propose a general framework for distribution-free nonparametric testing in multi-dimensions, based on a notion of “multivariate ranks” defined using the theory of optimal transport. We demonstrate the applicability of this approach by constructing exactly distribution-free tests for two classical nonparametric problems: (i) testing for the equality of two multivariate distributions, and (ii) testing for mutual independence between two random vectors. We investigate the consistency and asymptotic distributions of these tests, both under the null and local contiguous alternatives. We further study their local power and asymptotic (Pitman) efficiency, and show that a subclass of these tests achieve attractive efficiency lower bounds that mimic the remarkable efficiency results of Hodges and Lehmann (1956) and Chernoff and Savage (1958). To the best of our knowledge, these are the first collection of multivariate exactly distribution-free tests that provably achieve such attractive efficiency lower bounds.

Finally, we also study the rates of convergence of the estimated optimal transport maps, which are of pivotal importance in generative modeling, domain adaptation, etc. We will show that the natural plugin estimators for these maps achieve minimax optimal rates of convergence without any tuning parameters.

Alejandra Quintos Lima (Columbia Stats)

Title:  Dependent Stopping Times and an Application to Credit Risk Theory.

Abstract:  Stopping times are used in applications to model random arrivals. A standard assumption in many models is that the stopping times are conditionally independent, given an underlying filtration. This is a widely useful assumption, but there are circumstances where it seems to be unnecessarily strong. In the first part of the talk, we use a modified Cox construction, along with the bivariate exponential introduced by Marshall & Olkin (1967), to create a family of stopping times, which are not necessarily conditionally independent, allowing for a positive probability for them to be equal. We also present a series of results exploring the special properties of this construction.

In the second part of the talk, we present an application of our model to Credit Risk. We characterize the probability of a market failure which is defined as the default of two or more globally systemically important banks (G-SIBs) in a small interval of time. The default probabilities of the G-SIBs are correlated through the possible existence of a market-wide stress event. We derive various theorems related to market failure probabilities, such as the probability of a catastrophic market failure, the impact of increasing the number of G-SIBs in an economy, and the impact of changing the initial conditions of the economy’s state variables. We also show that if there are too many G-SIBs, a market failure is inevitable, i.e., the probability of a market failure tends to one as the number of G-SIBs tends to infinity.