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Causal Order Discovery based on Monotonic SCMs

Ali Izadi, Martin Ester

TL;DR

This work introduces a novel sequential procedure that directly identifies the causal order by iteratively detecting the root variable in monotonic Structural Causal Models, enabling the identification of a unique SCM without the need for multiple independence tests to break the Markov equivalence class.

Abstract

In this paper, we consider the problem of causal order discovery within the framework of monotonic Structural Causal Models (SCMs), which have gained attention for their potential to enable causal inference and causal discovery from observational data. While existing approaches either assume prior knowledge about the causal order or use complex optimization techniques to impose sparsity in the Jacobian of Triangular Monotonic Increasing maps, our work introduces a novel sequential procedure that directly identifies the causal order by iteratively detecting the root variable. This method eliminates the need for sparsity assumptions and the associated optimization challenges, enabling the identification of a unique SCM without the need for multiple independence tests to break the Markov equivalence class. We demonstrate the effectiveness of our approach in sequentially finding the root variable, comparing it to methods that maximize Jacobian sparsity.

Causal Order Discovery based on Monotonic SCMs

TL;DR

This work introduces a novel sequential procedure that directly identifies the causal order by iteratively detecting the root variable in monotonic Structural Causal Models, enabling the identification of a unique SCM without the need for multiple independence tests to break the Markov equivalence class.

Abstract

In this paper, we consider the problem of causal order discovery within the framework of monotonic Structural Causal Models (SCMs), which have gained attention for their potential to enable causal inference and causal discovery from observational data. While existing approaches either assume prior knowledge about the causal order or use complex optimization techniques to impose sparsity in the Jacobian of Triangular Monotonic Increasing maps, our work introduces a novel sequential procedure that directly identifies the causal order by iteratively detecting the root variable. This method eliminates the need for sparsity assumptions and the associated optimization challenges, enabling the identification of a unique SCM without the need for multiple independence tests to break the Markov equivalence class. We demonstrate the effectiveness of our approach in sequentially finding the root variable, comparing it to methods that maximize Jacobian sparsity.

Paper Structure

This paper contains 21 sections, 2 theorems, 8 equations, 2 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Background
  4. Monotonic Structural Causal Models
  5. Normalizing flows and TMI maps
  6. Proposed Method
  7. Problem definition
  8. Overview of the method
  9. Finding the root
  10. Causal order discovery
  11. Empirical evaluation
  12. Sequential or Permutation based Causal order discovery?
  13. Ablation Analysis
  14. Discussion
  15. Conclusion
  16. ...and 6 more sections

Key Result

Theorem 1

Suppose $X \in \mathcal{R}^{n\times d}$ is generated based on a monotonic SCM according to Equation eq:mscm. Then, for the multi-cause model eq:multi-cause, $\nabla_{X\backslash\{x_i\}}T_i(x_i, X\backslash\{x_i\})=\mathbf{\vec{0}}$, if and only if $x_i$ is a root in $\mathcal{G}$.

Figures (2)

  • Figure 1: Performance comparison of sequential order discovery vs permutation-based order discovery
  • Figure 2: Ablation analysis on the number of flow layers and hidden layers of MLP.

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2