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Efficiently Deciding Algebraic Equivalence of Bow-Free Acyclic Path Diagrams

Thijs van Ommen

TL;DR

This work targets causal discovery under latent confounding by focusing on bow-free acyclic path diagrams (BAPs) and the algebraic constraints inherent to linear structural equation models. It introduces graphically represented ideals and the crucial property of $I$-primary ideals to enable reliable model inclusion testing, then presents three randomized algorithms for testing a constraint, testing model inclusion, and testing model equivalence with one-sided error over a finite field. The authors prove correctness and provide complexity and error bounds, demonstrating that these methods efficiently distinguish algebraic models among BAPs and outperform empirical likelihood-based approaches. They also relate algebraic equivalence to other equivalence notions (distributional, Markov, nested Markov) and discuss practical implications, limitations, and future directions, including extensions beyond BAPs and connections to nested Markov theory.

Abstract

For causal discovery in the presence of latent confounders, constraints beyond conditional independences exist that can enable causal discovery algorithms to distinguish more pairs of graphs. Such constraints are not well-understood yet. In the setting of linear structural equation models without bows, we study algebraic constraints and argue that these provide the most fine-grained resolution achievable. We propose efficient algorithms that decide whether two graphs impose the same algebraic constraints, or whether the constraints imposed by one graph are a subset of those imposed by another graph.

Efficiently Deciding Algebraic Equivalence of Bow-Free Acyclic Path Diagrams

TL;DR

This work targets causal discovery under latent confounding by focusing on bow-free acyclic path diagrams (BAPs) and the algebraic constraints inherent to linear structural equation models. It introduces graphically represented ideals and the crucial property of -primary ideals to enable reliable model inclusion testing, then presents three randomized algorithms for testing a constraint, testing model inclusion, and testing model equivalence with one-sided error over a finite field. The authors prove correctness and provide complexity and error bounds, demonstrating that these methods efficiently distinguish algebraic models among BAPs and outperform empirical likelihood-based approaches. They also relate algebraic equivalence to other equivalence notions (distributional, Markov, nested Markov) and discuss practical implications, limitations, and future directions, including extensions beyond BAPs and connections to nested Markov theory.

Abstract

For causal discovery in the presence of latent confounders, constraints beyond conditional independences exist that can enable causal discovery algorithms to distinguish more pairs of graphs. Such constraints are not well-understood yet. In the setting of linear structural equation models without bows, we study algebraic constraints and argue that these provide the most fine-grained resolution achievable. We propose efficient algorithms that decide whether two graphs impose the same algebraic constraints, or whether the constraints imposed by one graph are a subset of those imposed by another graph.
Paper Structure (24 sections, 10 theorems, 13 equations, 3 figures, 1 table, 3 algorithms)

This paper contains 24 sections, 10 theorems, 13 equations, 3 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. The Graphically Represented Ideal
  5. $I$-Primary Ideals Enable Model Inclusion Testing
  6. Algorithms
  7. Testing a Constraint
  8. Testing Model Inclusion
  9. Testing Model Equivalence
  10. Other Equivalence Relations on Graphs
  11. Distributional Equivalence
  12. Graphical Conditions for Algebraic Equivalence
  13. Markov and Nested Markov Equivalence
  14. Experimental Results
  15. Discussion and Future Work
  16. ...and 9 more sections

Key Result

Theorem 1

Let $J$ be an $I$-primary ideal for $\overline{\mathcal{M}}(G')$. Let $\mathcal{M}(G)$ be another graphical model. Then $\mathcal{M}(G) \cap V(J) \setminus \overline{\mathcal{M}}(G')$ is of lower dimension than $\mathcal{M}(G)$.

Figures (3)

  • Figure 1: (a) A BAP for which the graphically represented ideal is $I$-primary but not PD-primary; (b) an ADMG for which the graphically represented ideal is not $I$-primary; (c) a BAP whose model may be mistakenly classified as a submodel of (b)'s model due to the latter's spurious components.
  • Figure 2: Two BAPs which are distributionally equivalent up to closure, but not distributionally equivalent, as $\mathcal{M}(G')$ excludes some covariance matrices that are present in $\mathcal{M}(G)$.
  • Figure 3: An algebraic equivalence class consisting of six BAPs. Graphs (a--c) differ by one edge (highlighted in yellow) and the same is true for (d--f). But between these two clusters, the difference is at least two edges (highlighted in pink).

Theorems & Definitions (22)

  • Example 1
  • Example 2
  • Theorem 1
  • Theorem 2
  • proof
  • Theorem 3
  • Lemma 4
  • Theorem 5
  • proof
  • Example 3
  • ...and 12 more