CAM: Causal additive models, high-dimensional order search and penalized regression (1310.1533v2)

Abstract: We develop estimation for potentially high-dimensional additive structural equation models. A key component of our approach is to decouple order search among the variables from feature or edge selection in a directed acyclic graph encoding the causal structure. We show that the former can be done with nonregularized (restricted) maximum likelihood estimation while the latter can be efficiently addressed using sparse regression techniques. Thus, we substantially simplify the problem of structure search and estimation for an important class of causal models. We establish consistency of the (restricted) maximum likelihood estimator for low- and high-dimensional scenarios, and we also allow for misspecification of the error distribution. Furthermore, we develop an efficient computational algorithm which can deal with many variables, and the new method's accuracy and performance is illustrated on simulated and real data.

• The paper decouples causal order search from feature selection by using a likelihood method for ordering and sparse regression for edge selection.
• It demonstrates consistent performance of maximum likelihood estimators in both low- and high-dimensional settings, even with misspecified error distributions.
• The efficient algorithm outperforms traditional methods like GES and PC in simulations and gene expression studies, reducing Structural Hamming and Intervention Distances.

Causal Additive Models: Estimation and Structure Learning

The paper "CAM: Causal Additive Models, High-Dimensional Order Search and Penalized Regression" by Bühlmann et al. presents a comprehensive approach to estimating and learning the structure of additive structural equation models (SEMs) in high-dimensional settings. The paper tackles the challenge of identifying causal relationships from observational data by introducing a framework that combines order search with penalized regression. This framework facilitates the estimation of potentially high-dimensional additive SEMs while addressing computational and identifiability challenges inherent in such models.

Key Contributions

1. Decoupling Order Search from Feature Selection: A significant contribution of this work is the decoupling of causal order search from feature or edge selection. The order search among variables is conducted using a (restricted) maximum likelihood estimation that does not rely on regularization, while the feature selection for the edges in a causal directed acyclic graph (DAG) employs sparse regression techniques. This separation simplifies the causal structure learning process significantly.
2. Consistency of Maximum Likelihood Estimators: The authors establish the consistency of the (restricted) maximum likelihood estimator across both low- and high-dimensional scenarios. This includes scenarios where the error distribution might be misspecified, thus broadening the applicability of their approach.
3. Efficient Computational Algorithm: The paper presents an efficient algorithm capable of handling many variables, which is crucial given the high-dimensional nature of the problems addressed. The method's performance is evaluated on both simulated and real data, demonstrating its accuracy and efficiency.
4. Empirical Validation: Through simulations and real gene expression data, the authors show that their approach yields more accurate estimations than traditional models, such as linear Gaussian SEMs, in cases where the models are identifiable from the available data.

Numerical Results and Implications

• Numerical Experiments: The algorithm's performance was compared against established methods like GES, PC, and RESIT. It consistently showed lower Structural Hamming Distance (SHD) and Structural Intervention Distance (SID), indicating more accurate causal inference under the nonlinear, high-dimensional settings it was designed for.
• Real Data Applications: The method was also applied to isoprenoid pathway gene expression data in Arabidopsis thaliana, showcasing its practical utility in uncovering biologically plausible causal relationships.

Theoretical and Practical Implications

• Theoretical Insights: The work underscores the importance of exploring additive models for causal inference, particularly in high-dimensional settings where traditional models fail due to identifiability and computational challenges.
• Practical Applications: This methodological framework is crucial for fields requiring robust causal inference from complex data, including genomics, neuroscience, and economics. It provides practical benefits in understanding causal mechanisms where nonlinear relationships play a key role.

Future Directions

Future research may explore extending this approach to models involving hidden variables, cyclic structures, or non-additive interactions. Additionally, integrating domain-specific knowledge into the causal inference process could enhance both the interpretability and reliability of the inferred graphs.

This paper thus represents a critical step forward in the field of causal discovery, providing a robust theoretical and computational framework for tackling high-dimensional causal inference problems.