Generalized Permutohedra from Probabilistic Graphical Models

TL;DR

This paper explores the geometric structures called generalized permutohedra derived from various probabilistic graphical models, providing new constructions and an algorithm for causal inference.

Contribution

It introduces a unified geometric framework for different graphical models using Minkowski sums of polytopes, extending the concept of graph associahedra.

Findings

Constructed polytopes for Gaussian and mixed graphical models.

Connected geometric structures to causal inference algorithms.

Extended the concept of graph associahedra to broader classes of models.

Abstract

A graphical model encodes conditional independence relations via the Markov properties. For an undirected graph these conditional independence relations can be represented by a simple polytope known as the graph associahedron, which can be constructed as a Minkowski sum of standard simplices. There is an analogous polytope for conditional independence relations coming from a regular Gaussian model, and it can be defined using multiinformation or relative entropy. For directed acyclic graphical models and also for mixed graphical models containing undirected, directed and bidirected edges, we give a construction of this polytope, up to equivalence of normal fans, as a Minkowski sum of matroid polytopes. Finally, we apply this geometric insight to construct a new ordering-based search algorithm for causal inference via directed acyclic graphical models.