## High-Dimensional Learning of Linear Causal Networks via Inverse Covariance Estimation

*Po-Ling Loh, Peter Bühlmann*; 15(Oct):3065−3105, 2014.

### Abstract

We establish a new framework for statistical estimation of
directed acyclic graphs (DAGs) when data are generated from a
linear, possibly non-Gaussian structural equation model. Our
framework consists of two parts: (1) inferring the moralized
graph from the support of the inverse covariance matrix; and (2)
selecting the best-scoring graph amongst DAGs that are
consistent with the moralized graph. We show that when the error
variances are known or estimated to close enough precision, the
true DAG is the unique minimizer of the score computed using the
reweighted squared $\ell_2$-loss. Our population-level results
have implications for the identifiability of linear SEMs when
the error covariances are specified up to a constant multiple.
On the statistical side, we establish rigorous conditions for
high-dimensional consistency of our two-part algorithm, defined
in terms of a "gap" between the true DAG and the next best
candidate. Finally, we demonstrate that dynamic programming may
be used to select the optimal DAG in linear time when the
treewidth of the moralized graph is bounded.

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