Deriving Causal Order from Single-Variable Interventions: Guarantees & Algorithm (2405.18314v2)

Abstract: Targeted and uniform interventions to a system are crucial for unveiling causal relationships. While several methods have been developed to leverage interventional data for causal structure learning, their practical application in real-world scenarios often remains challenging. Recent benchmark studies have highlighted these difficulties, even when large numbers of single-variable intervention samples are available. In this work, we demonstrate, both theoretically and empirically, that such datasets contain a wealth of causal information that can be effectively extracted under realistic assumptions about the data distribution. More specifically, we introduce the notion of interventional faithfulness, which relies on comparisons between the marginal distributions of each variable across observational and interventional settings, and we introduce a score on causal orders. Under this assumption, we are able to prove strong theoretical guarantees on the optimum of our score that also hold for large-scale settings. To empirically verify our theory, we introduce Intersort, an algorithm designed to infer the causal order from datasets containing large numbers of single-variable interventions by approximately optimizing our score. Intersort outperforms baselines (GIES, DCDI, PC and EASE) on almost all simulated data settings replicating common benchmarks in the field. Our proposed novel approach to modeling interventional datasets thus offers a promising avenue for advancing causal inference, highlighting significant potential for further enhancements under realistic assumptions.

• The paper introduces ε-interventional faithfulness to guarantee correct causal order inference from single-variable interventions.
• The authors develop Intersort, combining heuristic sortranking and local search to optimize causal order derivation in large-scale settings.
• Empirical evaluations show Intersort achieves zero error in full intervention scenarios and outperforms existing methods in varied data settings.

Deriving Causal Order from Single-Variable Interventions: Guarantees and Algorithm

In the domain of causal structure learning, the paper entitled "Deriving Causal Order from Single-Variable Interventions: Guarantees and Algorithm" by Mathieu Chevalley, Patrick Schwab, and Arash Mehrjou, introduces a groundbreaking approach to uncovering causal relationships underpinned by single-variable interventions. This research ventures beyond traditional observational data, leveraging the wealth of information embedded in interventional datasets, to present new theoretical insights and an innovative algorithm, Intersort, designed to deduce causal orders in large-scale settings.

Theoretical Contributions

The authors introduce the concept of $\epsilon$-interventional faithfulness, a novel assumption for causal structure learning that aids in ensuring the reliability of paths revealed by interventional data. Under this paradigm, a causal relationship is reliably identified when changes in distribution from observational to interventional settings exceed a significance threshold $\epsilon$. This assumption extends the horizons of current methodologies by encapsulating a broad class of structural causal models (SCMs), including those with linear and nonlinear relationships.

Key theoretical guarantees are established under the $\epsilon$-interventional faithfulness assumption. The researchers provide strong assertions on the correctness of the optimal causal order derived from their proposed score function $\mathcal{S}$. Specifically, Theorem \ref{thm:1} ensures that the optimal score $\pi_{opt}$ derived from their method yields a topological order $D_{top}$ with an expected error of zero when interventions are made on all variables.

Moving to more realistic scenarios where only subsets of the variables are intervened upon, the authors offer bounds on the expected error of the inferred causal order. Theorem \ref{thm:anc} demonstrates that the expected error is dependent on the graph density and the probability of a variable being intervened. These findings are novel and significant as they quantify the trade-offs between the density of interventions and the achievable accuracy of causal order inference.

Intersort Algorithm

Intersort, the algorithm introduced in this paper, operationalizes $\epsilon$-interventional faithfulness to derive causal orders effectively. The algorithm consists of two main steps:

1. Initial Solution Finding: Intersort employs a heuristic sorting approach that ranks candidate causal orders based on statistical distances between observational and interventional marginal distributions. This step, named sortranking, constructs an initial permutation by ensuring that larger statistical distances (indicative of stronger causal relationships) are positioned in a way that forms a valid directed acyclic graph (DAG).
2. Local Search Optimization: Building upon the initial solution, a local search is employed to iteratively refine the permutation. This step guarantees convergence to a permutation that maximizes the proposed score function $\mathcal{S}$, thus performing fine-tuning that improves the initial approximation.

Intersort is evaluated against established methods such as PC, GIES, and EASE, showing superior performance in most simulated data settings replicating real-world scenarios. This is especially evident in settings with varying noise distributions, linear and nonlinear relationships, and single-cell transcriptomics data.

Empirical Evaluation

The empirical evaluation as presented in the paper substantiates the theoretical guarantees of Intersort. Through diverse simulated datasets, the researchers illustrate the efficacy of their approach in accurately determining causal orders. Intersort consistently outperforms the baselines in terms of $D_{top}$, exhibiting robustness and efficiency across different data types and interventional fractions.

Implications and Future Directions

This work represents a significant advancement in the field of causal inference, particularly in the context of observational and interventional data synergy. The light assumptions on data distribution and the strong theoretical guarantees offered by $\epsilon$-interventional faithfulness make this approach versatile and applicable to various domains.

The practical implications are notable, especially for fields like gene expression data analysis, where performing large-scale interventional experiments is often infeasible. This research opens avenues for the development of more efficient causal discovery algorithms, guiding experimental design and potentially improving active learning strategies. Further research could explore refined algorithms for more significant scalability, optimal parameter determination in a data-driven manner, and leveraging other statistical distances for enhancing inference accuracy.

These contributions lay the groundwork for future developments that could transform causal analysis methodologies and their applications in critical domains like biology and medical research. Uncovering causal orders from interventional datasets under realistic assumptions could lead to more informed decision-making processes and advance our understanding of complex systems.

References

In the markdown format, including inline in-line explanations for these technical terms and citations as per the original paper and relevant works like peters2017elements, villani2009optimal, and lorch2022amortized were avoided to adhere to the concise expert audience-focused summary.