6.248 Graphical Models: A Geometric, Algebraic, and Combinatorial Perspective

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Graduate level
(Same subject as IDS.S21)
Units: 3-0-9
Prerequisites: linear algebra and probability (e.g. 18.06/18.700 and 6.041/6.431)
Schedule: MW11-12:30, room 1-135
Instructors:  Caroline Uhler and Elina Robeva
 
Description:
 
This subject will qualify as a Numerical Methods concentration subject.
 
In this research-oriented course we will introduce graphical models in the framework of exponential families.  We will see that polynomial equations and combinatorial constraints naturally arise and call for algebraic and combinatorial methods to advance the statistical methodology.
 
In particular, we will highlight the role of conic duality for Gaussian graphical models and polyhedral geometry for discrete graphical models.  We will also develop methods for causal inference making use of the inherent combinatorial and algebraic structure in directed graphical models.  Finally, we will discuss graphical models with hidden variables by highlighting the connections to tensor decompositions.
 
The overarching goal of this course is to provide an overview of the interplay of techniques from combinatorics, and applied algebraic geometry, with problems arising in statistics, in particular in graphical models.  Specific topics include exponential families, Grobner bases, conditional independence ideals, Bayesian networks, determinantal varieties, and hyperbolic polynomials.