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General Causal Imputation via Synthetic Interventions

Marco Jiralerspong, Thomas Jiralerspong, Vedant Shah, Dhanya Sridhar, Gauthier Gidel

TL;DR

A novel causal imputation estimator, generalized synthetic interventions (GSI), is introduced and it is proved the identifiability of this estimator for data generated from a more complex latent factor model.

Abstract

Given two sets of elements (such as cell types and drug compounds), researchers typically only have access to a limited subset of their interactions. The task of causal imputation involves using this subset to predict unobserved interactions. Squires et al. (2022) have proposed two estimators for this task based on the synthetic interventions (SI) estimator: SI-A (for actions) and SI-C (for contexts). We extend their work and introduce a novel causal imputation estimator, generalized synthetic interventions (GSI). We prove the identifiability of this estimator for data generated from a more complex latent factor model. On synthetic and real data we show empirically that it recovers or outperforms their estimators.

General Causal Imputation via Synthetic Interventions

TL;DR

A novel causal imputation estimator, generalized synthetic interventions (GSI), is introduced and it is proved the identifiability of this estimator for data generated from a more complex latent factor model.

Abstract

Given two sets of elements (such as cell types and drug compounds), researchers typically only have access to a limited subset of their interactions. The task of causal imputation involves using this subset to predict unobserved interactions. Squires et al. (2022) have proposed two estimators for this task based on the synthetic interventions (SI) estimator: SI-A (for actions) and SI-C (for contexts). We extend their work and introduce a novel causal imputation estimator, generalized synthetic interventions (GSI). We prove the identifiability of this estimator for data generated from a more complex latent factor model. On synthetic and real data we show empirically that it recovers or outperforms their estimators.

Paper Structure

This paper contains 15 sections, 1 theorem, 11 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our contributions:
  3. Problem Formulation
  4. Latent factor models
  5. Related Work
  6. Synthetic Controls (SC) (synthetic_controls)
  7. Synthetic Interventions (SI) (agarwal2023synthetic
  8. Synthetic Interventions - Actions (SI-A) (squires2022causal
  9. Matrix Completion (agarwal2021causal
  10. Method
  11. Theory
  12. Experiments & Results
  13. Synthetic data
  14. CMAP
  15. Conclusion

Key Result

Theorem 1

Suppose $\mathbf{M} \in \mathbb{R}^{N \times M \times D}$ satisfies the high-dimensional latent factor model (Assumption assu:multi_latent) with latent factors that satisfy Assumption assu:lin_comb. Then, if $\hat{\mathbf{y}}_{\text{test}}^{(d)}$ is derived as above, we have that $\forall d: \hat{\m

Figures (3)

  • Figure 1: Two-dimensional view of the incomplete matrix and the subsets used by SI-A/SI-C/GSI. The two last columns are missing either donor elements or the target and are thus not included in the train set.
  • Figure 2: Visualization of the full process for estimating $\textsc{GSI}(a_i, b_j)$. Steps 3 and 4 are performed for each dimension $d$ of the output independently.
  • Figure 3: Distribution of NRMSEs for various estimators on the test set ($n=578$). The bold values correspond to the median NRMSE for each estimator.

Theorems & Definitions (2)

  • Theorem 1
  • proof