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

Mathieu Chevalley, Patrick Schwab, Arash Mehrjou

TL;DR

This work tackles the problem of extracting causal order from datasets with many single-variable interventions. It introduces the $\epsilon$-interventional faithfulness$ assumption and a score-based framework that leads to Intersort, an algorithm consisting of an initialization step and a local search to maximize a causal-order score. The authors prove theoretical guarantees on the optimal score and provide finite-sample bounds for the top-order error, while empirically demonstrating that Intersort outperforms established baselines across diverse data-generating processes and remains robust to normalization. The results suggest that rich causal information is recoverable from interventional data under realistic assumptions, with potential practical impact on experimental design and causal inference in biology and related fields.

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 a novel variant 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.

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

TL;DR

This work tackles the problem of extracting causal order from datasets with many single-variable interventions. It introduces the -interventional faithfulness$ assumption and a score-based framework that leads to Intersort, an algorithm consisting of an initialization step and a local search to maximize a causal-order score. The authors prove theoretical guarantees on the optimal score and provide finite-sample bounds for the top-order error, while empirically demonstrating that Intersort outperforms established baselines across diverse data-generating processes and remains robust to normalization. The results suggest that rich causal information is recoverable from interventional data under realistic assumptions, with potential practical impact on experimental design and causal inference in biology and related fields.

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 a novel variant 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.
Paper Structure (39 sections, 11 theorems, 21 equations, 10 figures, 2 tables, 3 algorithms)

This paper contains 39 sections, 11 theorems, 21 equations, 10 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Causal ordering
  4. Causal Order: Definitions
  5. Statistical metric
  6. Distribution
  7. Causal Graph
  8. Causal Order
  9. Top Order Divergence ${D_{top}}$
  10. SCMs and interventions
  11. Distribution with Intervention $P_X^{(C,do(X_k := \Tilde{N}_k))}$
  12. $\epsilon$-Interventional faithfulness
  13. Latent confounders
  14. A score on causal orders
  15. Relaxation of $\epsilon$-Interventional Faithfulness
  16. ...and 24 more sections

Key Result

Lemma 1

Let $\pi^* \in \Pi^*$ be a causal ordering for $\mathcal{G}$. Then $D_{top} (\mathcal{G}, \pi^*) = 0$. rolland2022score

Figures (10)

  • Figure 1: Simulation and comparison between the two bounds, and between Intersort and the exact $\pi_{opt}$. For each setting, we draw $20$ graphs per setting, where a setting is the tuple $(p_{int}, p_{e})$. Then, for each graph, we run the algorithm on $10$ configurations, where each configuration corresponds to a draw of the targeted variables following $p_{int}$. For both $5$ and $30$ variables, we have $p_{int} \in \{0.25, 0.33, 0.5, 0.66, 0.75\}$. In the $5$ variables setting on the left, we have $p_{e} \in \{0.5, 0.66, 0.75\}$. In the $30$ variables settings on the right, we have $p_{e} \in \{0.05, 0.1, 0.2\}$. The settings are ordered on the x-axis following what we call the effective intervention ratio $\frac{p_{int}}{\sqrt{p_e}}$. We observe that the error is approximately monotonic when ordered by the effective intervention ratio.
  • Figure 2: Comparison of the performance of the baselines and of our model Intersort across diverse data domains as presented (linear, RFF, NN and GRN data), for $30$ variables. The x-axis corresponds to the fraction of variables that have been targeted by an intervention. The y-axis is the performance of causal ordering prediction as measured by the $D_{top}$ metric (see \ref{['def:dtop']}, lower is better). The violins are order from left to right: EASE, PC, DCDI, GIES, Intersort. Results for $10$ and $100$ variables can be found in the appendix (\ref{['fig:all-plots-app', 'fig:all-plots-100']}).
  • Figure 3: Simulation and comparison between the bounds of \ref{['thm:random_graph']} for scale-free networks, and between Intersort and the exact $\pi_{opt}$. For each setting, we draw $20$ graphs per setting following a Barabasi-Albert scale-free distribution, with average edge per variable in $\{1, 2, 3\}$. A setting is the tuple $(p_{int}, p_{e})$, where $p_{e} = \frac{2\mathrm{E}(\#edges)}{d(d-1)}$. Then, for each graph, we run the algorithm on $10$ configurations, where each configuration corresponds to a draw of the targeted variables following $p_{int}$. For both $5$ and $30$ variables, we have $p_{int} \in \{0.25, 0.33, 0.5, 0.66, 0.75\}$. The settings are ordered on the x-axis following what we call the effective intervention ratio $\frac{p_{int}}{\sqrt{p_e}}$. We observe that the error is approximately monotonic when ordered by the effective intervention ratio.
  • Figure 4: Simulation and comparison between the bounds of \ref{['thm:random_graph']} for scale-free networks and SORTRANKING. For each setting, we draw $2$ graphs per setting following a Barabasi-Albert scale-free distribution, with average edge per variable in $\{1, 2, 3\}$. A setting is the tuple $(p_{int}, p_{e})$, where $p_{e} = \frac{2\mathrm{E}(\#edges)}{d(d-1)}$. Then, for each graph, we run the algorithm on $1$ configuration, where each configuration corresponds to a draw of the targeted variables following $p_{int}$. For both $5$ and $30$ variables, we have $p_{int} \in \{0.25, 0.33, 0.5, 0.66, 0.75\}$. The settings are ordered on the x-axis following what we call the effective intervention ratio $\frac{p_{int}}{\sqrt{p_e}}$. Even though the objective can be approximately solved at scale, we observe room for improvement as the performance of SORTRANKING is above the upper-bound in many settings.
  • Figure 5: Simulation and comparison between the two bounds and SORTRANKING. For each setting, we draw $2$ graphs per setting, where a setting is the tuple $(p_{int}, p_{e})$. Then, for each graph, we run the algorithm on $1$ configurations, where each configuration corresponds to a draw of the targeted variables following $p_{int}$. For both $5$ and $30$ variables, we have $p_{int} \in \{0.25, 0.33, 0.5, 0.66, 0.75\}$. In the $1000$ variables setting on the left, we have $p_{e} \in \{0.005, 0.002, 0.001\}$. In the $20000$ variables settings on the right, we have $p_{e} \in \{0.0001, 0.00005, 0.00002\}$. The settings are ordered on the x-axis following what we call the effective intervention ratio $\frac{p_{int}}{\sqrt{p_e}}$. Even though the objective can be approximately solved at scale, we observe room for improvement as the performance of SORTRANKING is above the upper-bound in many settings.
  • ...and 5 more figures

Theorems & Definitions (20)

  • Lemma 1
  • Definition 1
  • Lemma 2
  • Lemma 3
  • Remark
  • Proposition 1
  • Theorem 1
  • Lemma 4
  • Theorem 2
  • Lemma 5
  • ...and 10 more