Advances in Learning Bayesian Networks of Bounded Treewidth

This work presents novel algorithms for learning Bayesian network structures with bounded treewidth. Both exact and approximate methods are developed. The exact method combines mixed-integer linear programming formulations for structure learning and treewidth computation. The approximate method consists in uniformly sampling $k$-trees (maximal graphs of treewidth $k$), and subsequently selecting, exactly or approximately, the best structure whose moral graph is a subgraph of that $k$-tree. Some properties of these methods are discussed and proven. The approaches are empirically compared to each other and to a state-of-the-art method for learning bounded treewidth structures on a collection of public data sets with up to 100 variables. The experiments show that our exact algorithm outperforms the state of the art, and that the approximate approach is fairly accurate.Comment: 23 pages, 2 figures, 3 table

Consistent Second-Order Conic Integer Programming for Learning Bayesian Networks

Bayesian Networks (BNs) represent conditional probability relations among a set of random variables (nodes) in the form of a directed acyclic graph (DAG), and have found diverse applications in knowledge discovery. We study the problem of learning the sparse DAG structure of a BN from continuous observational data. The central problem can be modeled as a mixed-integer program with an objective function composed of a convex quadratic loss function and a regularization penalty subject to linear constraints. The optimal solution to this mathematical program is known to have desirable statistical properties under certain conditions. However, the state-of-the-art optimization solvers are not able to obtain provably optimal solutions to the existing mathematical formulations for medium-size problems within reasonable computational times. To address this difficulty, we tackle the problem from both computational and statistical perspectives. On the one hand, we propose a concrete early stopping criterion to terminate the branch-and-bound process in order to obtain a near-optimal solution to the mixed-integer program, and establish the consistency of this approximate solution. On the other hand, we improve the existing formulations by replacing the linear "big-$M$" constraints that represent the relationship between the continuous and binary indicator variables with second-order conic constraints. Our numerical results demonstrate the effectiveness of the proposed approaches

Learning Bayesian Networks with the Saiyan algorithm

Some structure learning algorithms have proven to be effective in reconstructing hypothetical Bayesian Network (BN) graphs from synthetic data. However, in their mission to maximise a scoring function, many become conservative and minimise edges discovered. While simplicity is desired, the output is often a graph that consists of multiple independent graphical fragments or variables that do not enable full propagation of evidence. While this is not a problem in theory, it can be a problem in practice. This paper presents a novel unconventional heuristic local-search structure learning algorithm, called Saiyan, which returns a directed acyclic graph that enables full propagation of evidence. Forcing the algorithm to connect all data variables and to direct all of the edges discovered implies that the additional forced arcs are not expected to be correct at the rate of those identified unrestrictedly, and this evidently has a negative impact on the evaluation score of the discovered graph. Still, based on both synthetic and real-world experiments, the Saiyan algorithm demonstrates competitive performance relative to other state-of-the-art constraint-based, score-based, and hybrid structure learning algorithms

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

Towards using the chordal graph polytope in learning decomposable models

The motivation for this paper is the integer linear programming approach to learning the structure of a decomposable graphical model. We have chosen to represent decomposable models by means of special zero–one 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. In this theoretical paper, we introduce a class of clutter inequalities (valid for the vectors in the polytope) and show that all of them are facet-defining for the polytope. We dare to conjecture that they lead to a complete polyhedral description of the polytope. Finally, we propose a linear programming method to solve the separation problem with these inequalities for the use in a cutting plane approach