We consider the problem of estimating the differences between two causal directed acyclic graph (DAG) models given i.i.d. samples from each model. This is of interest for example in genomics, where large-scale gene expression data is becoming available under different cellular contexts, for different cell types, or disease states. Changes in the structure or edge weights of the underlying causal graphs reflect alterations in the gene regulatory networks and provide important insights into the emergence of a particular phenotype. While the individual networks are usually very large, containing high-degree hub nodes and thus difficult to learn, the overall change between two related networks can be sparse. We here provide the first provably consistent method for directly estimating the differences in a pair of causal DAGs without separately learning two possibly large and dense DAG models and computing their difference. Our two-step algorithm first uses invariance tests between regression coefficients of the two data sets to estimate the skeleton of the difference graph and then orients some of the edges using invariance tests between regression residual variances. We demonstrate the properties of our method through a simulation study and apply it to the analysis of gene expression data from ovarian cancer and during T-cell activation.
@cite_19: 
@cite_37: We propose a new method for detecting changes in Markov network structure between two sets of samples. Instead of naively fitting two Markov network models separately to the two data sets and figuring out their difference, we directly learn the network structure change by estimating the ratio of Markov network models. This density-ratio formulation naturally allows us to introduce sparsity in the network structure change, which highly contributes to enhancing interpretability. Furthermore, computation of the normalization term, a critical bottleneck of the naive approach, can be remarkably mitigated. We also give the dual formulation of the optimization problem, which further reduces the computation cost for large-scale Markov networks. Through experiments, we demonstrate the usefulness of our method.
@cite_34: It is often of interest to understand how the structure of a genetic network differs between two conditions. In this paper, each condition-specific network is modelled using the precision matrix of a multivariate normal random vector, and a method is proposed to directly estimate the difference of the precision matrices. In contrast to other approaches, such as separate or joint estimation of the individual matrices, direct estimation does not require those matrices to be sparse, and thus can allow the individual networks to contain hub nodes. Under the assumption that the true differential network is sparse, the direct estimator is shown to be consistent in support recovery and estimation. It is also shown to outperform existing methods in simulations, and its properties are illustrated on gene expression data from late-stage ovarian cancer patients.
A simpler task is learning differences of undirected graphical models. Let @math and @math denote the precision matrices corresponding to @math and @math . The support of @math consists of the edges in the undirected graph (UG) models corresponding to @math . We define the difference-UG ( D-UG ) by @math , with @math if and only if @math for @math . Two recent methods that directly learn the difference of two UG models are KLIEP @cite_37 and DPM @cite_34 ; for a review and comparison of these methods see @cite_19 . These methods can be used as a first step towards estimating the D-DAG @math : under genericity assumptions, the formulae for @math in imply that if @math then @math . Hence, the skeleton of @math is a subgraph of @math , i.e., @math . In the following section we present our algorithm showing how to obtain @math and determine some of the edge directions in @math . We end this section with a piece of notation needed for introducing our algorithm; we define the to be @math .