Greedy Causal Discovery Is Geometric

Abstract

The purpose of this paper is twofold. We investigate a simple necessary condition, called the rhombus criterion, for two vertices in a polytope not to form an edge and show that in many examples of 0/1-polytopes it is also sufficient. We explain how also when this is not the case, the criterion can give a good algorithm for determining the edges of high-dimenional polytopes.In particular we study the Chordal graph polytope, which arises in the theory of causality and is an important example of a characteristic imset polytope. We prove that, asymptotically, for almost all pairs of vertices the rhombus criterion holds. We conjecture it to hold for all pairs of vertices.

Diameters of the Characteristic Imset Polytopes

“…To do this we interpret i → j in G to mean i being a direct cause of j. Studený, Hemmecke, and Lindner transformed this question into a linear program over CIM n [16,18]. The authors of [9] showed that the edge structure of CIM n is of particular interest. In their paper we are given a geometric interpretation of several greedy algorithms as edge-walks over CIM n and its faces.…”

“…In their paper we are given a geometric interpretation of several greedy algorithms as edge-walks over CIM n and its faces. In particular, for any undirected graph G, the face [9] (of CIM n ) is studied. A complete characterisation of the edges of CIM G when G is a tree or a cycle was recently discovered [8].…”

“…In particular, for any undirected graph G, the face [9] (of CIM n ) is studied. A complete characterisation of the edges of CIM G when G is a tree or a cycle was recently discovered [8]. For general G, less is known and the only edges with a clear interpretation are the ones given in [9,Proposition 3.2], namely that changing the direction of a single edge in a DAG, that does not create a directed cycle, gives us an edge in CIM G (see Theorem 2.1).…”

“…In particular, in Section 2 we see that for a general undirected graph G = ([n], E) we have diam CIM G ≤ |E|. If G is a tree we can, via the work of [8], improve this bound to diam CIM G ≤ n − 2 and give a lower bound in terms of the maximal path length of G. This is done in Section 2.1. Finally in Section 2.2 we show, using a new type of edge, that diam CIM n ≤ 2n − 2.…”

“…. , X n , find the MEC that best encodes the observed conditional independence statements in D. This has often been interpreted as finding the DAG (or MEC of DAGs) maximising BIC(G, D), where BIC denotes the Bayesian information criterion [4,8,9,19]. Importantly the BIC is score equivalent and decomposable [4], that is, it is a linear function over CIM n .…”