The characteristic imset polytope of Bayesian networks with ordered nodes

TL;DR

This paper studies the geometric structure of characteristic imset polytopes for Bayesian networks with ordered nodes, revealing their composition as products of simplices and describing their edges and facets.

Contribution

It characterizes the structure of cim-polytopes for ordered Bayesian networks, including diagnosis models, and generalizes to networks with fixed node orderings.

Findings

Cim-polytopes for diagnosis models are direct products of simplices.

Complete combinatorial descriptions of edges and facets are provided.

Results extend to Bayesian networks with fixed node orderings.

Abstract

In 2010, M. Studen\'y, R. Hemmecke, and S. Linder explored a new algebraic description of graphical models, called characteristic imsets. Compare with standard imsets, characteristic imsets have several advantages: they are still unique vector representative of conditional independence structures, they are 0-1 vectors, and they are more intuitive in terms of graphs than standard imsets. After defining a characteristic imset polytope (cim-polytope) as the convex hull of all characteristic imsets with a given set of nodes, they also showed that a model selection in graphical models, which maximizes a quality criterion, can be converted into a linear programming problem over the cim-polytope. However, in general, for a fixed set of nodes, the cim-polytope can have exponentially many vertices over an exponentially high dimension. Therefore, in this paper, we focus on the family of directed acyclic graphs (DAGs) whose nodes have a fixed order. This family includes diagnosis models which can be described by Bipartite graphs with a set of $m$ nodes and a set of $n$ nodes for any $m, n \in \Z_+$. In this paper, we first consider cim-polytopes for all diagnosis models and show that these polytopes are direct products of simplices. Then we give a combinatorial description of all edges and all facets of these polytopes. Finally, we generalize these results to the cim-polytopes for all Bayesian networks with a fixed underlying ordering of nodes with or without fixed (or forbidden) edges.