[Paper Review] A Bayesian Approach to Constraint Based Causal Inference

Tom Claassen, Tom Heskes|arXiv (Cornell University)|Oct 16, 2012

Bayesian Modeling and Causal Inference28 references38 citations

TL;DR

This paper proposes a Bayesian Constraint-based Causal Discovery (BCCD) algorithm that integrates Bayesian scoring with constraint-based causal inference to improve robustness and accuracy in finite-sample settings. By ranking and processing constraint decisions by posterior probability, BCCD reduces error propagation and provides reliable causal structures with uncertainty quantification, outperforming FCI and Conservative PC in empirical tests while identifying high-confidence causal edges.

ABSTRACT

We target the problem of accuracy and robustness in causal inference from finite data sets. Some state-of-the-art algorithms produce clear output complete with solid theoretical guarantees but are susceptible to propagating erroneous decisions, while others are very adept at handling and representing uncertainty, but need to rely on undesirable assumptions. Our aim is to combine the inherent robustness of the Bayesian approach with the theoretical strength and clarity of constraint-based methods. We use a Bayesian score to obtain probability estimates on the input statements used in a constraint-based procedure. These are subsequently processed in decreasing order of reliability, letting more reliable decisions take precedence in case of con icts, until a single output model is obtained. Tests show that a basic implementation of the resulting Bayesian Constraint-based Causal Discovery (BCCD) algorithm already outperforms established procedures such as FCI and Conservative PC. It can also indicate which causal decisions in the output have high reliability and which do not.

Motivation & Objective

• To address the trade-off between robustness and theoretical rigor in causal inference from limited data.
• To reduce error propagation in constraint-based methods by incorporating uncertainty estimates from Bayesian scoring.
• To produce a causal structure with quantified reliability, distinguishing high-confidence from low-confidence edges.
• To combine the theoretical clarity of constraint-based methods with the uncertainty-awareness of Bayesian approaches.
• To develop a method that outperforms established algorithms like FCI and Conservative PC in accuracy and reliability.

Proposed method

• The method uses a Bayesian score to compute posterior probabilities for each conditional independence statement in the constraint-based procedure.
• Constraint decisions are ranked by their posterior probability, with higher-probability statements processed first to minimize error propagation.
• The algorithm processes constraints in decreasing order of reliability, resolving conflicts by prioritizing more certain decisions.
• A single, consistent causal model is constructed by iteratively applying the most reliable constraints first.
• The final output includes both the causal structure and a reliability score for each edge, indicating confidence in its inclusion.
• The approach is implemented as a basic BCCD algorithm and evaluated on synthetic and real-world data.

Experimental results

Research questions

• RQ1 Can Bayesian scoring improve the robustness of constraint-based causal discovery in finite-sample settings?
• RQ2 How does prioritizing constraint decisions by posterior probability affect the accuracy and reliability of the resulting causal structure?
• RQ3 Can the proposed method outperform established algorithms like FCI and Conservative PC in terms of structural accuracy?
• RQ4 To what extent can the method quantify uncertainty in individual causal edges?
• RQ5 Does the integration of Bayesian scoring with constraint-based inference lead to more reliable causal discovery than either approach alone?

Key findings

• The BCCD algorithm outperforms both FCI and Conservative PC in terms of structural accuracy on benchmark datasets.
• The method successfully reduces error propagation by prioritizing high-confidence constraint decisions over less reliable ones.
• BCCD provides a reliability score for each edge, enabling users to distinguish between high-confidence and uncertain causal relationships.
• Even a basic implementation of BCCD achieves superior performance compared to state-of-the-art constraint-based methods.
• The integration of Bayesian scoring with constraint-based inference results in a more robust and informative causal discovery process.
• The algorithm demonstrates improved performance in finite-sample regimes where traditional constraint-based methods are prone to error.