15 research outputs found
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
The Chordal Graph Polytope for Learning Decomposable Models
This theoretical paper is inspired by an integer linear programming (ILP) approach to learning the structure of decomposable models. We intend to represent decomposable models by special zeroone vectors, named characteristic imsets. Our approach leads to the study of a special polytope, defined as the convex hull of all characteristic imsets for chordal graphs, named the chordal graph polytope. We introduce a class of clutter inequalities and show that all of them are valid for (the vectors in) the polytope. In fact, these inequalities are even facet-defining for the polytope and we dare to conjecture that they lead to a complete polyhedral description of the polytope. Finally, we propose an LP method to solve the separation problem with these inequalities for use in a cutting plane approach
Polytopes Arising from Binary Multi-way Contingency Tables and Characteristic Imsets for Bayesian Networks
The main theme of this dissertation is the study of polytopes arising from binary multi-way contingency tables and characteristic imsets for Bayesian networks.
Firstly, we study on three-way tables whose entries are independent Bernoulli ran- dom variables with canonical parameters under no three-way interaction generalized linear models. Here, we use the sequential importance sampling (SIS) method with the conditional Poisson (CP) distribution to sample binary three-way tables with the sufficient statistics, i.e., all two-way marginal sums, fixed. Compared with Monte Carlo Markov Chain (MCMC) approach with a Markov basis (MB), SIS procedure has the advantage that it does not require expensive or prohibitive pre-computations. Note that this problem can also be considered as estimating the number of lattice points inside the polytope defined by the zero-one and two-way marginal constraints. The theorems in Chapter 2 give the parameters for the CP distribution on each column when it is sampled. In this chapter, we also present the algorithms, the simulation results, and the results for Samson’s monks data.
Bayesian networks, a part of the family of probabilistic graphical models, are widely applied in many areas and much work has been done in model selections for Bayesian networks. The second part of this dissertation investigates the problem of finding the optimal graph by using characteristic imsets, where characteristic imsets are defined as 0-1 vector representations of Bayesian networks which are unique up to Markov equivalence. Characteristic imset polytopes are defined as the convex hull of all characteristic imsets we consider. It was proven that the problem of finding optimal Bayesian network for a specific dataset can be converted to a linear programming problem over the characteristic imset polytope [51]. In Chapter 3, we first consider characteristic imset polytopes for all diagnosis models and show that these polytopes are direct product of simplices. Then we give the combinatorial description of all edges and all facets of these polytopes. At the end of this chapter, we generalize these results to the characteristic imset polytopes for all Bayesian networks with a fixed underlying ordering of nodes.
Chapter 4 includes discussion and future work on these two topics
Polyhedral aspects of score equivalence in Bayesian network structure learning
This paper deals with faces and facets of the family-variable polytope and the characteristic-imset polytope, which are special polytopes used in integer linear programming approaches to statistically learn Bayesian network structure. A common form of linear objectives to be maximized in this area leads to the concept of score equivalence (SE), both for linear objectives and for faces of the family-variable polytope. We characterize the linear space of SE objectives and establish a one-to-one correspondence between SE faces of the family-variable polytope, the faces of the characteristic-imset polytope, and standardized supermodular functions. The characterization of SE facets in terms of extremality of the corresponding supermodular function gives an elegant method to verify whether an inequality is SE-facet-defining for the family-variable polytope. We also show that when maximizing an SE objective one can eliminate linear constraints of the family-variable polytope that correspond to non-SE facets. However, we show that solely considering SE facets is not enough as a counter-example shows; one has to consider the linear inequality constraints that correspond to facets of the characteristic-imset polytope despite the fact that they may not define facets in the family-variable mode
Greedy Causal Discovery is Geometric
Finding a directed acyclic graph (DAG) that best encodes the conditional
independence statements observable from data is a central question within
causality. Algorithms that greedily transform one candidate DAG into another
given a fixed set of moves have been particularly successful, for example the
GES, GIES, and MMHC algorithms. In 2010, Studen\'y, Hemmecke and Lindner
introduced the characteristic imset polytope, , whose vertices
correspond to Markov equivalence classes, as a way of transforming causal
discovery into a linear optimization problem. We show that the moves of the
aforementioned algorithms are included within classes of edges of
and that restrictions placed on the skeleton of the candidate
DAGs correspond to faces of . Thus, we observe that GES, GIES,
and MMHC all have geometric realizations as greedy edge-walks along
. Furthermore, the identified edges of
strictly generalize the moves of these algorithms. Exploiting this
generalization, we introduce a greedy simplex-type algorithm called greedy CIM,
and a hybrid variant, skeletal greedy CIM, that outperforms current competitors
among hybrid and constraint-based algorithms.Comment: 35 pages, 8 figure
On the Edges of Characteristic Imset Polytopes
The edges of the characteristic imset polytope, , were
recently shown to have strong connections to causal discovery as many
algorithms could be interpreted as greedy restricted edge-walks, even though
only a strict subset of the edges are known. To better understand the general
edge structure of the polytope we describe the edge structure of faces with a
clear combinatorial interpretation: for any undirected graph we have the
face , the convex hull of the characteristic imsets of
DAGs with skeleton . We give a full edge-description of
when is a tree, leading to interesting connections
to other polytopes. In particular the well-studied stable set polytope can be
recovered as a face of when is a tree. Building on
this connection we are also able to give a description of all edges of
when is a cycle, suggesting possible inroads for
generalization. We then introduce an algorithm for learning directed trees from
data, utilizing our newly discovered edges, that outperforms classical methods
on simulated Gaussian data.Comment: 36 pages, 5 figure
Toric Ideals of Characteristic Imsets via Quasi-Independence Gluing
Characteristic imsets are 0-1 vectors which correspond to Markov equivalence
classes of directed acyclic graphs. The study of their convex hull, named the
characteristic imset polytope, has led to new and interesting geometric
perspectives on the important problem of causal discovery. In this paper we
begin the study of the associated toric ideal. We develop a new generalization
of the toric fiber product, which we call a quasi-independence gluing, and show
that under certain combinatorial homogeneity conditions, one can iteratively
compute a Gr\"obner basis via lifting. For faces of the characteristic imset
polytope associated to trees, we apply this technique to compute a Gr\"obner
basis for the associated toric ideal. We end with a study of the characteristic
ideal of the cycle and propose directions for future work.Comment: 19 pages, 7 figure