Characterizations of Decomposable Dependency Models

Decomposable dependency models possess a number of interesting and useful properties. This paper presents new characterizations of decomposable models in terms of independence relationships, which are obtained by adding a single axiom to the well-known set characterizing dependency models that are isomorphic to undirected graphs. We also briefly discuss a potential application of our results to the problem of learning graphical models from data.Comment: See http://www.jair.org/ for any accompanying file

Learning Score-Optimal Chordal Markov Networks via Branch and Bound

Graphical models are commonly used to encode conditional independence assumptions between random variables. Here we focus on undirected graphical models called chordal Markov networks. Specifically, we will consider the chordal Markov network structure learning problem (CMSL), where the aim is to find (or "learn") a graph structure that best fits the given data with respect to a given decomposable scoring function. We introduce a branch and bound search algorithm for CMSL which represents chordal Markov network structures as decomposable DAGs. We show how revisiting equivalent solution candidates can be avoided in the search by detecting symmetries among graph structures. For the symmetry breaking we apply specific rules by van Beek and Hoffman (CP 2015), and also propose a new rule that takes advantage of the special nature of decomposable DAGs. In addition, we show how we can achieve on-the-fly score pruning for CMSL. We also propose methods for obtaining strong upper bounds for CMSL that help us close branches in the search tree. We implement a dynamic programming algorithm to find the optimal Bayesian network structures and then use the scores of those graphs as upper bounds. We also show how we can relax the requirement for decomposability in decomposable DAGs in order to achieve even stronger upper bounds. Furthermore, we propose a method for obtaining an initial lower bound in CMSL by turning a Bayesian network structure into a chordal Markov network structure. Empirically we show that our approach is competitive with the recently proposed CMSL algorithms by being able to sometimes scale up to 20 variables within 24 hours with unbounded treewidth. We also report that our branch and bound requires considerably less memory than the fastest of the recently proposed algorithms for CMSL

Markov models for fMRI correlation structure: is brain functional connectivity small world, or decomposable into networks?

Correlations in the signal observed via functional Magnetic Resonance Imaging (fMRI), are expected to reveal the interactions in the underlying neural populations through hemodynamic response. In particular, they highlight distributed set of mutually correlated regions that correspond to brain networks related to different cognitive functions. Yet graph-theoretical studies of neural connections give a different picture: that of a highly integrated system with small-world properties: local clustering but with short pathways across the complete structure. We examine the conditional independence properties of the fMRI signal, i.e. its Markov structure, to find realistic assumptions on the connectivity structure that are required to explain the observed functional connectivity. In particular we seek a decomposition of the Markov structure into segregated functional networks using decomposable graphs: a set of strongly-connected and partially overlapping cliques. We introduce a new method to efficiently extract such cliques on a large, strongly-connected graph. We compare methods learning different graph structures from functional connectivity by testing the goodness of fit of the model they learn on new data. We find that summarizing the structure as strongly-connected networks can give a good description only for very large and overlapping networks. These results highlight that Markov models are good tools to identify the structure of brain connectivity from fMRI signals, but for this purpose they must reflect the small-world properties of the underlying neural systems