Standard imsets for undirected and chain graphical models

We derive standard imsets for undirected graphical models and chain graphical models. Standard imsets for undirected graphical models are described in terms of minimal triangulations for maximal prime subgraphs of the undirected graphs. For describing standard imsets for chain graphical models, we first define a triangulation of a chain graph. We then use the triangulation to generalize our results for the undirected graphs to chain graphs.Comment: Published at http://dx.doi.org/10.3150/14-BEJ611 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Generalized Permutohedra from Probabilistic Graphical Models

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.Comment: Appendix B is expanded. Final version to appear in SIAM J. Discrete Mat

The Characteristic Imset Polytope of Bayesian Networks with Ordered Nodes

In 2010, M. Studený, R. Hemmecke, and S. Lindner explored a new algebraic description of graphical models, called characteristic imsets. Compared with standard imsets, characteristic imsets have several advantages: they are still unique vector representatives of conditional independence structures, 0-1 vectors, and 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 whose nodes have a fixed order. This family includes diagnosis models described by bipartite graphs with a set of m nodes and a set of n nodes for any m, n ∈ Z+. 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

A geometric view on learning Bayesian network structures

We recall the basic idea of an algebraic approach to learning Bayesian network (BN) structures, namely to represent every BN structure by a certain (uniquely determined) vector, called a standard imset. The main result of the paper is that the set of standard imsets is the set of vertices ( = extreme points) of a certain polytope. Motivated by the geometric view, we introduce the concept of the geometric neighborhood for standard imsets, and, consequently, for BN structures. Then we show that it always includes the inclusion neighborhood, which was introduced earlier in connection with the greedy equivalence search (GES) algorithm. The third result is that the global optimum of an affine function over the polytope coincides with the local optimum relative to the geometric neighborhood. To illustrate the new concept by an example, we describe the geometric neighborhood in the case of three variables and show it differs from the inclusion neighborhood. This leads to a simple example of the failure of the GES algorithm if data are not “generated ” from a perfectly Markovian distribution. The point is that one can avoid this failure if the search technique is based on the geometric neighborhood instead. We also found out what is the geometric neighborhood in the case of four and five variables