Order-independent constraint-based causal structure learning (1211.3295v2)

Abstract: We consider constraint-based methods for causal structure learning, such as the PC-, FCI-, RFCI- and CCD- algorithms (Spirtes et al. (2000, 1993), Richardson (1996), Colombo et al. (2012), Claassen et al. (2013)). The first step of all these algorithms consists of the PC-algorithm. This algorithm is known to be order-dependent, in the sense that the output can depend on the order in which the variables are given. This order-dependence is a minor issue in low-dimensional settings. We show, however, that it can be very pronounced in high-dimensional settings, where it can lead to highly variable results. We propose several modifications of the PC-algorithm (and hence also of the other algorithms) that remove part or all of this order-dependence. All proposed modifications are consistent in high-dimensional settings under the same conditions as their original counterparts. We compare the PC-, FCI-, and RFCI-algorithms and their modifications in simulation studies and on a yeast gene expression data set. We show that our modifications yield similar performance in low-dimensional settings and improved performance in high-dimensional settings. All software is implemented in the R-package pcalg.

• The paper introduces PC-stable variants that remove order-dependence in skeleton estimation for more reliable causal graph learning.
• It refines v-structure identification with Conservative and Majority Rule approaches to ensure consistent edge orientation.
• Empirical results on simulations and yeast gene expression data demonstrate improved true discovery rates and reduced structural errors.

Order-independent Constraint-based Causal Structure Learning

The paper by Colombo and Maathuis addresses a significant concern in the field of constraint-based methods for causal structure learning, particularly focusing on the well-known PC-algorithm and its variants like FCI, RFCI, and CCD. The primary issue under scrutiny is the order-dependence of these algorithms, whereby the sequence of input variables can influence the resulting causal graph. This order-dependence is often negligible in low-dimensional contexts but becomes problematic in high-dimensional scenarios, leading to highly variable outputs.

Core Contributions

The paper's essential advancements lie in proposing modifications to the PC-algorithm that eliminate this order-dependence without sacrificing the high-dimensional consistency features inherent in the original algorithm:

1. PC-stable Algorithm: By adapting the adjacency set used during the skeleton estimation phase, the authors introduce the PC-stable algorithm. This modification ensures that edge deletions do not affect other conditional independence tests at the same level, making the skeleton estimation order-independent.
2. V-structure Identification: The authors present two approaches, the Conservative PC (CPC) and Majority Rule PC (MPC), to make the v-structure determination order-independent. Both methods rely on the assessment of all possible separating sets to judge triple ambiguities, with MPC offering a less conservative alternative.
3. Full Order-independence: Extending upon PC-stable, the LCPC and LMPC variants utilize lists for candidate v-structures and orientation rules, allowing bi-directional edge representations to address ambiguities during the orientation phase fully.
4. Adaptations to FCI and RFCI: These modifications extend to FCI and RFCI, producing stable variants such as FCI-stable, augmenting them with conservative or majority rule orientations as needed. These implementations maintain order-independence across skeleton, v-structure, and orientation phases.

Empirical Validation

The authors conduct an extensive empirical analysis involving simulations and a real-world yeast gene expression dataset. The results demonstrate:

• Simulations: In high-dimensional settings, order-independent algorithms notably reduce structural errors (SHD/SHD edge marks) and variances compared to their order-dependent counterparts. Particularly, the PC-stable and RFCI-stable versions achieve better true discovery rates (TDR) for skeleton estimation.
• Yeast Dataset: For the complex yeast gene expression data, PC-stable provides a consistently sparse skeleton, capturing stable edges across different orderings. The simulations further highlight improvements in the stability and accuracy of inferred causal effects.

Theoretical Implications and Future Directions

The theoretical implications are profound, suggesting that ensuring order-independence can lead to more reliable causal inferences, especially for datasets with high dimensionality. This advancement holds potential for a wide array of applications in fields requiring causal discovery and inference. Future research might explore:

• Scalability: Further enhancing computational efficiency, given PC-stable tends to perform more tests.
• Refinement of Orientation Rules: Addressing order-dependence in the more intricate orientation processes of algorithms like FCI with potentially new heuristic methods.
• Causal Interpretation: Refining the interpretation of bi-directed edges to enhance practical causal analysis.

The modifications proposed by Colombo and Maathuis represent a significant step forward in constraint-based causal learning. By ensuring that the outputs are invariant to the order of input variables, these methods provide a more reliable tool for researchers working with high-dimensional causal discovery tasks. The implementations in the R-package pcalg reflect the practical applicability of this research.