Conference Agenda

Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).

 
Session Overview
Session
MS172, part 2: Algebraic statistics
Time:
Wednesday, 10/Jul/2019:
10:00am - 12:00pm

Location: Unitobler, F-121
52 seats, 100m^2

Presentations
10:00am - 12:00pm

Algebraic Statistics

Chair(s): Jose Israel Rodriguez (UW Madison), Elizabeth Gross (University of Hawaiʻi at Mānoa)

Algebraic statistics studies statistical models through the lens of algebra, geometry, and combinatorics. From model selection to inference, this interdisciplinary field has seen applications in a wide range of statistical procedures. This session will focus broadly on new developments in algebraic statistics, both on the theoretical side and the applied side.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Geometry of Exponential Graph Models

Ha Khanh Nguyen
The Ohio State University

When given network data, we can either compute descriptive statistics (degree distribution, diameter, clustering co- efficient, etc.) or we can find a model that explains the data. Modeling allow us to test hypotheses about edge formation, understand the uncertainty associated with the observed outcomes, and conduct inferences about whether the network substructures are more commonly observed than by chance. Modeling is also used for simulation and assessment of local effects. Exponential random graph models (ERGMs) are families of distributions defined by a set of network statistics and, thus, give rise to interesting graph theoretic questions. Our research focuses on the ERGMs where the edge, 2-path, and triangle counts are the sufficient statistics. These models are useful for modeling networks with a transitivity effect such as social networks. One of the most popular research questions for statisticians is the goodness-of-fit testing, how well does the model ”fit” the data? This is a difficult question for ERGMs. And one way to answer this question is to understand the reference set. Given an observed network G, the reference set of G is the set of simple graphs with the same edge, 2-path, and triangle counts as G. In algebraic geometry, it is called the fiber of G and are the 0-1 points on an algebraic variety, which we refer to as the reference variety. The goal of this paper is to understand reference variety through the lens of algebraic geometry.

 

Moment Varieties of Measures on Polytopes

Kathlén Kohn
University of Oslo

This talk brings many areas together: discrete geometry, statistics, algebraic geometry, invariant theory, geometric modeling, symbolic and numerical computations. We study the algebraic relations among moments of uniform probability distributions on polytopes. This is already a non-trivial matter for quadrangles in the plane. In fact, we need to combine invariant theory of the affine group with numerical algebraic geometry to compute first relevant relations. Moreover, the numerator of the generating function of all moments of a fixed polytope is the adjoint of the polytope, which is known from geometric modeling. We prove the conjecture that the adjoint is the unique polynomial of minimal degree which vanishes on the non-faces of a simple polytope. This talk is based on joint work with Kristian Ranestad, Boris Shapiro and Bernd Sturmfels.

 

The stratification of the maximum likelihood degree for toric varieties

Serkan Hosten
SFSU

The lattice points of a lattice polytope give rise to a family of toric varieties when we allow complex coefficients in the monomial parametrization of the "usual" toric variety associated to the polytope. The maximum likelihood degree (ML degree) of any member of this family is at most the normalized volume of the polytope. The set of coefficient vectors associated to ML degrees smaller than the volume is parametrized by Gelfand-Kapranov-Zelevinsky's principal A-determinant. Not much is known about how the ML degree changes as one moves in the parameter space. We will discuss what we know starting with toric surfaces.

 

Nested Determinantal Constraints in Linear Structural Equation Models

Elina Robeva
MIT

Directed graphical models specify noisy functional relationships among a collection of random variables. In the Gaussian case, each such model corresponds to a semi-algebraic set of positive definite covariance matrices. The set is given via parametrization, and much work has gone into obtaining an implicit description in terms of polynomial (in-)equalities. Implicit descriptions shed light on problems such as parameter identification, model equivalence, and constraint-based statistical inference. For models given by directed acyclic graphs, which represent settings where all relevant variables are observed, there is a complete theory: All conditional independence relations can be found via graphical d-separation and are sufficient for an implicit description. The situation is far more complicated, however, when some of the variables are hidden. We consider models associated to mixed graphs that capture the effects of hidden variables through correlated error terms. The notion of trek separation explains when the covariance matrix in such a model has submatrices of low rank and generalizes d-separation. However, in many cases, such as the infamous Verma graph, the polynomials defining the graphical model are not determinantal, and hence cannot be explained by d-separation or trek-separation. We show that these constraints often correspond to the vanishing of nested determinants and can be graphically explained by a notion of restricted trek separation.