Graphical continuous Lyapunov models

The linear Lyapunov equation of a covariance matrix parametrizes the equilibrium covariance matrix of a stochastic process. This parametrization can be interpreted as a new graphical model class, and we show how the model class behaves under marginalization and introduce a method for structure learning via $\ell_1$-penalized loss minimization. Our proposed method is demonstrated to outperform alternative structure learning algorithms in a simulation study, and we illustrate its application for protein phosphorylation network reconstruction.Comment: 10 pages, 5 figure

Finding Non-overlapping Clusters for Generalized Inference Over Graphical Models

Graphical models use graphs to compactly capture stochastic dependencies amongst a collection of random variables. Inference over graphical models corresponds to finding marginal probability distributions given joint probability distributions. In general, this is computationally intractable, which has led to a quest for finding efficient approximate inference algorithms. We propose a framework for generalized inference over graphical models that can be used as a wrapper for improving the estimates of approximate inference algorithms. Instead of applying an inference algorithm to the original graph, we apply the inference algorithm to a block-graph, defined as a graph in which the nodes are non-overlapping clusters of nodes from the original graph. This results in marginal estimates of a cluster of nodes, which we further marginalize to get the marginal estimates of each node. Our proposed block-graph construction algorithm is simple, efficient, and motivated by the observation that approximate inference is more accurate on graphs with longer cycles. We present extensive numerical simulations that illustrate our block-graph framework with a variety of inference algorithms (e.g., those in the libDAI software package). These simulations show the improvements provided by our framework.Comment: Extended the previous version to include extensive numerical simulations. See http://www.ima.umn.edu/~dvats/GeneralizedInference.html for code and dat

On the causal interpretation of acyclic mixed graphs under multivariate normality

In multivariate statistics, acyclic mixed graphs with directed and bidirected edges are widely used for compact representation of dependence structures that can arise in the presence of hidden (i.e., latent or unobserved) variables. Indeed, under multivariate normality, every mixed graph corresponds to a set of covariance matrices that contains as a full-dimensional subset the covariance matrices associated with a causally interpretable acyclic digraph. This digraph generally has some of its nodes corresponding to hidden variables. We seek to clarify for which mixed graphs there exists an acyclic digraph whose hidden variable model coincides with the mixed graph model. Restricting to the tractable setting of chain graphs and multivariate normality, we show that decomposability of the bidirected part of the chain graph is necessary and sufficient for equality between the mixed graph model and some hidden variable model given by an acyclic digraph

Recent results on well-balanced orientations

In this paper we consider problems related to Nash-Williams´ Strong Orientation Theorem and Odd-Vertex Pairing Theorem. These theorems date to 1960 and up to now not much is known about their relationship to other subjects in graph theory. We investigated many approaches to find a more transparent proof for these theorems and possibly generalizations of them. In many cases we found negative answers: counter-examples and NP-completeness results. For example we show that the weighted and the degree-constrained versions of the well-balanced orientation problem are NP-hard. We also show that it is NP-hard to find a minimum cost feasible odd-vertex pairing or to decide whether two graphs with some common edges have simultaneous well-balanced orientations or not. Nash-Williams´ original approach was to define best-balanced orientations with feasible odd-vertex pairings: we show here that not every best-balanced orientation can be obtained this way. However we prove that in the global case this is true: every smooth k-arc-connected orientation can be obtained through a k-feasible odd-vertex pairing. The aim of this paper is to help to find a transparent proof for the Strong Orientation Theorem. In order to achieve this we propose some other approaches and raise some open questions, too. (c) 2008 Elsevier B.V. All rights reserved