Algebraic implicitization techniques in the graphical models program

Abstract

Many aspects of daily life now rely on technology driven by advanced techniques for analyzing multivariate data. Graphical models provide the general framework within which much of these analyses take place. However, different data types require different graphical models, with each requiring its own basic mathematical theory to guide sound statistical inference. Developing this theory for a family of graphical models requires solutions to fundamental questions, where emerging mathematical approaches are utilizing what can collectively be called algebraic implicitization techniques. These notes provide an introduction to the basics of graphical models and the algebraic implicitization techniques now appearing in instances of the graphical models program.

This is a preprint of a chapter intended for publication in a Springer Lecture Notes in Mathematics series based on the lectures given at the 43rd Finnish Summer School on Probability and Statistics in Lammi, Finland. It surveys basic results on graphical models and algebraic techniques, with the intent of being broadly accessible