10:00am - 12:00pmAlgebraic 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)
Testing model fit for networks: algebraic statistics of mixture models and beyond
Sonja Petrovic
IIT
We consider statistical models for relational data that can be represented as a network. The nodes in the network are individuals, organizations, proteins, neurons, or brain regions, while edges---directed or undirected--- specific types of relationships between the nodes such as personal or organizational affinities or other social/financial relationships, or some physical or functional links such as co-activation of brain regions. One of the key open problems in this area is testing whether a proposed statistical model fits the data at hand. Algebraic statistics is known to provide theoretically reliable tools for testing model fit for a class of models that are log-linear exponential families; let's call these log-linear ERGMs. In this talk, we will discuss how the machinery can be extended to mixtures of log-linear ERGMs and other general linear exponential-family models that need not be log-linear, and what the hurdles are that need to be overcome in order for this set of tools to be generalizable, scalable and practical.
Oriented Gaussoids
Thomas Kahle
OvGU Magdeburg
An oriented gaussoid is a combinatorial structure that captures the possible signs of correlations among Gaussian random variables. We introduce this concept and present approaches to the classification and construction of oriented gaussoids, drawing parallels to oriented matroids, which capture the possible signs of dependencies in linear algebra.
Ideals of Gaussian Graphical Models
Seth Sullivant
NCSU
Gaussian graphical models are semialgebraic subsets of the cone of positive definite matrices. We will report on recent results trying to characterize the vanishing ideals of these models, in particular situations where they are generated by determinantal constraints.
Combinatorial matrix theory in structural equation models
Marc Harkonen
Georgia Tech
Many operations on matrices can be viewed from a combinatorial point of view by considering graphs associated to the matrix. For example, the determinant and inverse of a matrix can be computed from the linear subgraphs and 1-connections of the Coates digraph associated to the matrix. This combinatorial approach also naturally takes advantage of the sparsity structure of the matrix, which makes it ideal for applications in linear structural equation models. Another advantage of these combinatorial methods is the fact that they are often agnostic on whether the mixed graph contains cycles. As an example, we obtain a symbolic representation of the entries of the covariance matrix as a finite sum. In general, this sum will become similar to the well known trek rule, but where each half of the trek is a 1-connection instead of a path. This method of computing the covariance matrix can be easily implemented in computer algebra systems, and scales extremely well when the mixed graph has few cycles.