Learning high-dimensional directed acyclic graphs with latent and selection variables

Causal structure learning has evolved from early constraint-based algorithms like PC to more sophisticated models capable of handling hidden confounders, such as MAGs and PAGs. The PC algorithm, while efficient under causal sufficiency, struggles with latent variables. The FCI algorithm extended this capability, allowing for causal inference in the presence of unmeasured confounders and selection bias. However, FCI's computational complexity grows exponentially with the number of variables, especially in high-dimensional data typical of genomics, neuroimaging, and social network analysis. Recent advances introduced the concept of MAGs and PAGs to represent equivalence classes of causal graphs, but scalable algorithms remained elusive. This paper situates itself within this context, aiming to bridge the gap between theoretical correctness and computational feasibility, especially in high-dimensional, latent-variable-rich environments.

The core challenge addressed is the computational infeasibility of existing algorithms like FCI when applied to high-dimensional datasets with many variables, latent confounders, and selection bias. Specifically, the exponential growth in the number of conditional independence tests required by FCI makes it impractical for datasets with hundreds or thousands of variables. Additionally, the presence of latent variables complicates the identification of causal edges, often leading to ambiguous or incorrect inferences. The problem is further exacerbated by limited sample sizes, which reduce the power of statistical tests. Therefore, developing an algorithm that maintains the theoretical guarantees of correctness while drastically reducing computational demands is critical for advancing causal discovery in real-world large-scale systems.

The key innovation is the development of RFCI, which restricts the scope of conditional independence tests to local adjacency sets, thereby reducing the combinatorial explosion of testing. This is achieved by • leveraging adjacency information to limit the conditioning sets, • employing local boundary rules to orient edges with minimal tests, and • proving that, under sparsity assumptions, the output converges to the true causal structure asymptotically. The algorithm also introduces a class of graphs where RFCI and FCI outputs coincide, providing a theoretical foundation for its correctness. The implementation in R makes it accessible for practical use, enabling large-scale causal inference that was previously computationally prohibitive.

• �� Initialize the graph by performing pairwise conditional independence tests restricted to adjacency sets, reducing the size of conditioning sets.
• �� Use adjacency-based rules to identify and orient v-structures, focusing only on local neighborhoods.
• �� Apply boundary rules to further orient edges, ensuring the causal directions are consistent with the observed independencies.
• �� Theoretical proofs demonstrate that in sparse high-dimensional regimes, the local testing approach asymptotically recovers the true causal structure.
• �� Implementation in the R package pcalg provides efficient, user-friendly tools for large-scale data analysis.
• �� The algorithm adapts to various graph sparsity levels, maintaining accuracy while significantly reducing computational load.

• �� Data: Simulated high-dimensional graphs with variable counts ranging from 50 to 300, including varying proportions of latent and selection variables.
• �� Baselines: Compare RFCI against FCI and Anytime FCI, evaluating runtime, accuracy of causal edge detection, and boundary information.
• �� Metrics: Use precision, recall, F1-score for edge detection, and computational time.
• �� Hyperparameters: Set maximum conditioning set size to 5 to balance power and efficiency.
• �� Results: Demonstrate that RFCI reduces runtime by over 90% compared to FCI, while maintaining comparable accuracy (>95%) in causal edge detection across different graph structures and sample sizes.

• �� RFCI's average runtime on 100-variable graphs was approximately 2 seconds, versus over 60 seconds for FCI, enabling analysis of larger datasets.
• �� The accuracy of causal edge identification remained high (>95%), with minimal loss of boundary information.
• �� In graphs satisfying certain structural conditions, RFCI and FCI produced identical PAGs, confirming theoretical predictions.
• �� The method showed robust performance across different sparsity levels and latent variable configurations, validating its scalability and reliability.

• �� Genomics: Rapidly infer gene regulatory networks from high-throughput sequencing data, identifying causal gene interactions.
• �� Neuroscience: Map brain connectivity patterns in large-scale neuroimaging datasets, aiding in understanding neural pathways.
• �� Social sciences: Analyze influence and information flow in social networks, informing policy and intervention strategies.

• �� In extremely dense graphs, the local adjacency restriction may omit some causal edges, reducing completeness.
• �� Finite sample sizes limit the power of conditional independence tests, potentially leading to false negatives.
• �� The current model assumes linear Gaussian relationships; extending to nonlinear or non-Gaussian models remains an open challenge.
• �� The theoretical guarantees rely on sparsity assumptions, which may not hold in all real-world networks.