High-dimensional learning of linear causal networks via inverse covariance estimation (1311.3492v1)

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 l_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.

• The paper proposes a two-step algorithm that uses inverse covariance estimation to infer a network's moralized graph before selecting the optimal causal structure with a score function.
• Theoretical proofs show that the method can identify the true linear causal structure even with non-Gaussian errors or misspecified error variances under certain conditions.
• The approach employs techniques like Graphical Lasso and dynamic programming, making it statistically consistent and computationally efficient for sparse high-dimensional data.

Statistical Estimation of Linear Causal Networks

The paper "Linear SEMs via Inverse Covariance Estimation" by Po-Ling Loh and Peter Bühlmann establishes a novel framework for statistical estimation in high-dimensional causal networks, particularly those characterized by directed acyclic graphs (DAGs) resulting from a linear structural equation model (SEM). The authors introduce a two-part algorithmic strategy focused on leveraging inverse covariance matrix estimation to infer the moralized graph of a DAG and subsequently search for the optimal causal structure.

Methodological Overview

The paper's central methodological contribution lies in two sequential steps:

1. Inverse Covariance Estimation to Infer Moralized Graph: By estimating the inverse covariance matrix of joint data, the authors infer the moralized graph. This process is validated, extending beyond Gaussian constraints, allowing applications in non-Gaussian linear SEMs. It asserts that for most linear coefficients, the support of the inverse covariance matrix aligns with the moralized graph's edge structure.
2. Optimal DAG Selection with Score Functions: Within the DAGs compatible with the moralized graph, the optimal structure is identified using a score function, specifically a reweighted squared l2-loss. The authors propose dynamic programming methods to compute this efficiently when the moralized graph's treewidth is limited.

Theoretical Insights

The authors present rigorous proofs confirming that the squared l2-loss uniquely minimizes at the true DAG in scenarios where error variances are known or sufficiently approximated. It implies identifiability of linear SEMs despite the homogeneous nature of error variances, encompassing models with both Gaussian and non-Gaussian errors.

Additionally, statistical consistency of the proposed algorithm is demonstrated through high-dimensional statistics, weakened from previous stringent conditions, thus better accommodating scenarios with increased complexity and node numbers.

Numerical and Statistical Implications

The paper outlines conditions under which the method retains consistency despite misspecified error variances, crucial for practical applications. It establishes bounds concerning variance ratio deviations required to ensure algorithmic robustness in model selection.

Moreover, graphical Lasso techniques are advocated for inverse covariance matrix estimation, with statistical guarantees provided for both low-dimensional and high-dimensional settings. This positions the algorithm as notably efficient computationally when dealing with sparse datasets.

Computational Considerations

Integrating dynamic programming allows computationally efficient determination of the best-scoring DAG within the constraints of bounded treewidth, extending applicability to larger network scales. The paper advances existing benchmarks in causal inference, aligning efficiency with precision through constrained DAG spaces informed by inverse covariance estimation.

Conclusion and Future Work

Loh and Bühlmann's framework offers substantial improvements in identifying linear causal structures by coupling the theoretical robustness of inverse covariance matrix relationships with the practicality of efficient algorithms. Future research is envisaged to further refine gap parameter characterization for diverse graphical structures and explore scoring function optimizations for uncertain error variances, enhancing the adaptability of this framework to broader classes of structural models.

This foundational work paves the way for more agile handling of high-dimensional causal network estimation, with implications extending across multiple domains requiring precise statistical network identification.