  We consider the problem of estimating the differences between two causal
directed acyclic graph (DAG) models with a shared topological order given
i.i.d. samples from each model. This is of interest for example in genomics,
where changes in the structure or edge weights of the underlying causal graphs
reflect alterations in the gene regulatory networks. 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.
