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Distinguishing Cause from Effect with Causal Velocity Models (2502.05122v2)

Published 7 Feb 2025 in stat.ML, cs.LG, and stat.ME

Abstract: Bivariate structural causal models (SCM) are often used to infer causal direction by examining their goodness-of-fit under restricted model classes. In this paper, we describe a parametrization of bivariate SCMs in terms of a causal velocity by viewing the cause variable as time in a dynamical system. The velocity implicitly defines counterfactual curves via the solution of initial value problems where the observation specifies the initial condition. Using tools from measure transport, we obtain a unique correspondence between SCMs and the score function of the generated distribution via its causal velocity. Based on this, we derive an objective function that directly regresses the velocity against the score function, the latter of which can be estimated non-parametrically from observational data. We use this to develop a method for bivariate causal discovery that extends beyond known model classes such as additive or location scale noise, and that requires no assumptions on the noise distributions. When the score is estimated well, the objective is also useful for detecting model non-identifiability and misspecification. We present positive results in simulation and benchmark experiments where many existing methods fail, and perform ablation studies to examine the method's sensitivity to accurate score estimation.

Summary

  • The paper introduces a novel framework that parameterizes structural causal models using causal velocities, generalizing beyond traditional noise assumptions.
  • It derives a score-based objective function that nonparametrically estimates causal velocities from observational data, reducing risks of model misspecification.
  • The method is validated through simulations and benchmarks, demonstrating improved bivariate causal discovery in challenging, assumption-heavy scenarios.

An Overview of Causal Velocity Models for Bivariate Causal Discovery

The paper presents a novel approach to bivariate causal discovery by introducing causal velocity models, which offer a new parametrization of structural causal models (SCMs) through causal velocity functions. Unlike traditional approaches, this framework views the causal mechanism as a dynamical system where the cause variable acts as time. This perspective allows the derivation of counterfactual outcomes as solutions to initial value problems. Through measure transport theories, the manuscript establishes a unique link between SCMs and score functions of generated distributions via their causal velocities.

Key Contributions

  1. Parametrization via Causal Velocity: The paper proposes to parameterize SCMs through causal velocities that describe infinitesimal counterfactuals. This approach generalizes beyond existing model assumptions such as additive noise models (ANMs) and location-scale noise models (LSNMs).
  2. Score Function Estimation: The paper derives an objective function that regresses causal velocities against score functions, estimated nonparametrically from observational data. This regression circumvents the need for noise distribution assumptions, traditionally required in likelihood-based methods, reducing the risk of model misspecification.
  3. Bivariate Causal Discovery Method: By leveraging the derived score-based objective function, the authors introduce a method for bivariate causal discovery that is robust to model misspecifications, augmenting the current approaches limited by constraints on noise or model classes.
  4. Experimental Validation and Simulations: The manuscript reports simulation and benchmark experiments that demonstrate superiority over existing methods, particularly highlighting scenarios where traditional methods fail due to strong underlying assumptions.

Theoretical Implications

This framework extends the theoretical foundation of causal inference by providing a tool to explore causal dynamics without the restrictive dependencies imposed by predefined noise distributions. The method's foundation is in leveraging the continuity equation that uniquely ties score functions to causal velocities.

Practical Implications and Future Directions

The proposed method has substantial implications for practical applications where causal inference from purely observational data is critical. In domains such as econometrics, epidemiology, and artificial intelligence, the flexibility to identify causal directions without stringent model assumptions can significantly enhance decision-making processes.

Future work could expand this paradigm to multivariate settings and explore the integration with high-dimensional data scenarios, potentially amplifying the reach and utility of AI systems in causative reasoning tasks. Additionally, improved methods for score estimation to enhance finite-sample behavior could strengthen applications in limited-data environments common in real-world issues.

In conclusion, the paper provides a significant methodological advancement in causal discovery, driven by the integration of dynamical systems theory with modern causal inference challenges. The use of score-based evaluations opens new avenues for robust inference, potentially impacting various scientific pursuits reliant on causal explanations.

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