Generalized Permutohedra from Probabilistic Graphical Models
Author(s)
Mohammadi, Fatemeh; Uhler, Caroline; Wang, Charles; Yu, Josephine
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© 2018 Society for Industrial and Applied Mathematics. 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.
Date issued
2018Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science; Massachusetts Institute of Technology. Institute for Data, Systems, and SocietyJournal
SIAM Journal on Discrete Mathematics
Publisher
Society for Industrial & Applied Mathematics (SIAM)