Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Learning Unknown Intervention Targets in Structural Causal Models from Heterogeneous Data

Yuqin Yang, Saber Salehkaleybar, Negar Kiyavash

Abstract

We study the problem of identifying the unknown intervention targets in structural causal models where we have access to heterogeneous data collected from multiple environments. The unknown intervention targets are the set of endogenous variables whose corresponding exogenous noises change across the environments. We propose a two-phase approach which in the first phase recovers the exogenous noises corresponding to unknown intervention targets whose distributions have changed across environments. In the second phase, the recovered noises are matched with the corresponding endogenous variables. For the recovery phase, we provide sufficient conditions for learning these exogenous noises up to some component-wise invertible transformation. For the matching phase, under the causal sufficiency assumption, we show that the proposed method uniquely identifies the intervention targets. In the presence of latent confounders, the intervention targets among the observed variables cannot be determined uniquely. We provide a candidate intervention target set which is a superset of the true intervention targets. Our approach improves upon the state of the art as the returned candidate set is always a subset of the target set returned by previous work. Moreover, we do not require restrictive assumptions such as linearity of the causal model or performing invariance tests to learn whether a distribution is changing across environments which could be highly sample inefficient. Our experimental results show the effectiveness of our proposed algorithm in practice.

Learning Unknown Intervention Targets in Structural Causal Models from Heterogeneous Data

Abstract

We study the problem of identifying the unknown intervention targets in structural causal models where we have access to heterogeneous data collected from multiple environments. The unknown intervention targets are the set of endogenous variables whose corresponding exogenous noises change across the environments. We propose a two-phase approach which in the first phase recovers the exogenous noises corresponding to unknown intervention targets whose distributions have changed across environments. In the second phase, the recovered noises are matched with the corresponding endogenous variables. For the recovery phase, we provide sufficient conditions for learning these exogenous noises up to some component-wise invertible transformation. For the matching phase, under the causal sufficiency assumption, we show that the proposed method uniquely identifies the intervention targets. In the presence of latent confounders, the intervention targets among the observed variables cannot be determined uniquely. We provide a candidate intervention target set which is a superset of the true intervention targets. Our approach improves upon the state of the art as the returned candidate set is always a subset of the target set returned by previous work. Moreover, we do not require restrictive assumptions such as linearity of the causal model or performing invariance tests to learn whether a distribution is changing across environments which could be highly sample inefficient. Our experimental results show the effectiveness of our proposed algorithm in practice.
Paper Structure (35 sections, 6 theorems, 19 equations, 8 figures, 2 tables, 3 algorithms)

This paper contains 35 sections, 6 theorems, 19 equations, 8 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Methodology
  4. Problem definition
  5. Recovery Phase
  6. Matching Phase
  7. T-faithfulness assumption
  8. Matching phase under causal sufficiency
  9. Matching phase in the presence of latent confounders
  10. Experiments
  11. Comparison with previous work
  12. Conclusions
  13. Further discussion on the recovery phase
  14. Detailed Description about Contrastive Learning Approach and Nonlinear ICA
  15. How constrastive learning approach above is applied in our work.
  16. ...and 20 more sections

Key Result

Proposition 1

Assume that $\min(D-1,|{\bf O}|)\geq|{\bf T}|$ (Recall that $D$ is the number of environments and ${\bf O}$ and ${\bf L}$ are the set of observed and latent variables in the system, respectively). By utilizing the contrastive-learning approach, the exogenous noises in ${\bf N}_{{\bf T}}$ can be reco

Figures (8)

  • Figure 1: (a) The causal graph of the SCM considered in Example \ref{['example:Th2']}. (b) The corresponding auxiliary graph according to Definition \ref{['def:auxillary_graph']}. (c) The MAG of the augmented graph defined in jaber2020causal, which indicates the output of their algorithm. (d) The causal graph of an alternative SCM that has the same auxiliary graph.
  • Figure 2: Comparison of the recovery outputs when latent confounders are not in ${\bf T}$. The dashed circle represents the theoretical limitation of the recovery of ${\bf T}$. It may include observed variables that are theoretically indistinguishable from the true intervention targets due to the presence of latent confounders.
  • Figure 3: Comparison of LIT algorithm with previous work in locating intervention targets.
  • Figure 4: An example of SCM: Intervention targets ${\bf T}=\{X_1,X_2,$$X_5\}$ are shown by red circles.
  • Figure 5: Performance of the algorithms under Setting (1) (Linear Gaussian model under causal sufficiency assumption). Error bars represents the 25% and 75% percentiles of the corresponding metrics. Note that for $D=32$, PreDITEr algorithm (orange line) overlaps with UT-IGSP algorithm (green line) and CITE algorithm (red line) in recall, as both algorithms have 1.0 recall for all $n$.
  • ...and 3 more figures

Theorems & Definitions (22)

  • Proposition 1
  • Remark 1
  • Proposition 2
  • Theorem 1
  • Remark 2
  • Proposition 3
  • Example 1
  • Definition 1: Auxiliary graph
  • Theorem 2
  • Remark 3
  • ...and 12 more