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Signature Kernel Conditional Independence Tests in Causal Discovery for Stochastic Processes

Georg Manten, Cecilia Casolo, Emilio Ferrucci, Søren Wengel Mogensen, Cristopher Salvi, Niki Kilbertus

TL;DR

This paper develops conditional independence (CI) constraints on coordinate processes over selected intervals that are Markov with respect to the acyclic dependence graph (allowing self-loops) induced by a general SDE model and proposes a flexible, consistent signature kernel-based CI test to infer these constraints from data.

Abstract

Inferring the causal structure underlying stochastic dynamical systems from observational data holds great promise in domains ranging from science and health to finance. Such processes can often be accurately modeled via stochastic differential equations (SDEs), which naturally imply causal relationships via "which variables enter the differential of which other variables". In this paper, we develop conditional independence (CI) constraints on coordinate processes over selected intervals that are Markov with respect to the acyclic dependence graph (allowing self-loops) induced by a general SDE model. We then provide a sound and complete causal discovery algorithm, capable of handling both fully and partially observed data, and uniquely recovering the underlying or induced ancestral graph by exploiting time directionality assuming a CI oracle. Finally, to make our algorithm practically usable, we also propose a flexible, consistent signature kernel-based CI test to infer these constraints from data. We extensively benchmark the CI test in isolation and as part of our causal discovery algorithms, outperforming existing approaches in SDE models and beyond.

Signature Kernel Conditional Independence Tests in Causal Discovery for Stochastic Processes

TL;DR

This paper develops conditional independence (CI) constraints on coordinate processes over selected intervals that are Markov with respect to the acyclic dependence graph (allowing self-loops) induced by a general SDE model and proposes a flexible, consistent signature kernel-based CI test to infer these constraints from data.

Abstract

Inferring the causal structure underlying stochastic dynamical systems from observational data holds great promise in domains ranging from science and health to finance. Such processes can often be accurately modeled via stochastic differential equations (SDEs), which naturally imply causal relationships via "which variables enter the differential of which other variables". In this paper, we develop conditional independence (CI) constraints on coordinate processes over selected intervals that are Markov with respect to the acyclic dependence graph (allowing self-loops) induced by a general SDE model. We then provide a sound and complete causal discovery algorithm, capable of handling both fully and partially observed data, and uniquely recovering the underlying or induced ancestral graph by exploiting time directionality assuming a CI oracle. Finally, to make our algorithm practically usable, we also propose a flexible, consistent signature kernel-based CI test to infer these constraints from data. We extensively benchmark the CI test in isolation and as part of our causal discovery algorithms, outperforming existing approaches in SDE models and beyond.
Paper Structure (43 sections, 7 theorems, 43 equations, 9 figures, 10 tables, 3 algorithms)

This paper contains 43 sections, 7 theorems, 43 equations, 9 figures, 10 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background and Related Work
  3. Methodology
  4. Causal Discovery in the Acyclic SDE Model
  5. Signature Kernel Conditional Independence Test
  6. Experiments
  7. Conclusion and Future Work
  8. Proofs and Theoretical Digressions
  9. Summary of Notations
  10. Intuition and Details for the SDE Model
  11. Counterexample for Cyclic Causal Discovery in the SDE Model
  12. Signature Kernel Details
  13. Additional mathematical details.
  14. Intuition.
  15. Testing on Time Points
  16. ...and 28 more sections

Key Result

Proposition 3.1

Assume the dependence graph $\mathcal{G}$ of the SDE model is acyclic except for loops and associate $X^i_{[0,s]}$ with $i_0\in V_0$ and $X^i_{[s,s+h]}$ with $i_1 \in V_1$. Then $\tilde{\mathcal{G}}$ is acyclic and the independence relation $\mathop{\mathrm{\perp\space\perp}}\nolimits_{s,h}^+$ satis

Figures (9)

  • Figure 1: Illustration of causal discovery in the SDE model (A), leveraging conditional independencies in the observed samples (B) to infer the dependence graph of the SDE (C).
  • Figure 1: Causal discovery for acyclic SDEs.
  • Figure 2: The lifted dependence graph $\tilde{\mathcal{G}}$ (right) for a DAG $\mathcal{G}$ (left) with colors highlighting the correspondence of selected edges.
  • Figure 3: Test power for $X^1_{[0,t]} \mathop{\mathrm{\perp\space\perp}}\nolimits X^2_{[0,t]}$ over $\tfrac{a_{21}}{a_{22}}$. Lines (shades) are means (standard errors) over $1000$ SDE instances.
  • Figure 4: An example graph demonstrating the advantage of our approach under partial observations.
  • ...and 4 more figures

Theorems & Definitions (21)

  • Proposition 3.1: Markov property
  • Theorem 3.2
  • Corollary 3.3: post-processing
  • Example A.1
  • proof : Proof of \ref{['prop:Markov']}
  • Remark A.2
  • Remark A.3
  • Remark A.4
  • proof : Proof of \ref{['thm:causalDisc']}
  • Definition A.5: (Global) Faithfulness of $\mathop{\mathrm{\perp\space\perp}}\nolimits_{s,h}^+$ in $\tilde{\mathcal{G}}$
  • ...and 11 more