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Flow IV: Counterfactual Inference In Nonseparable Outcome Models Using Instrumental Variables

Marc Braun, Jose M. Peña, Adel Daoud

Abstract

To reach human level intelligence, learning algorithms need to incorporate causal reasoning. But identifying causality, and particularly counterfactual reasoning, remains elusive. In this paper, we make progress on counterfactual inference in nonseparable outcome models by utilizing instrumental variables (IVs). IVs are a classic tool for mitigating bias from unobserved confounders when estimating causal effects. While IV methods for effect estimation have been extended to nonseparable outcome models under different assumptions, existing IV approaches to counterfactual prediction typically assume one-dimensional outcomes and additive noise. In this paper, we show that under standard IV assumptions, along with the assumption that the outcome function is invertible and has a triangular structure, then the treatment-outcome relationship becomes identifiable from observed data. We furthermore propose a method to learn the outcome function utilizing normalizing flows. This outcome function estimator can then be used to perform counterfactual inference. We refer to the method as Flow IV.

Flow IV: Counterfactual Inference In Nonseparable Outcome Models Using Instrumental Variables

Abstract

To reach human level intelligence, learning algorithms need to incorporate causal reasoning. But identifying causality, and particularly counterfactual reasoning, remains elusive. In this paper, we make progress on counterfactual inference in nonseparable outcome models by utilizing instrumental variables (IVs). IVs are a classic tool for mitigating bias from unobserved confounders when estimating causal effects. While IV methods for effect estimation have been extended to nonseparable outcome models under different assumptions, existing IV approaches to counterfactual prediction typically assume one-dimensional outcomes and additive noise. In this paper, we show that under standard IV assumptions, along with the assumption that the outcome function is invertible and has a triangular structure, then the treatment-outcome relationship becomes identifiable from observed data. We furthermore propose a method to learn the outcome function utilizing normalizing flows. This outcome function estimator can then be used to perform counterfactual inference. We refer to the method as Flow IV.

Paper Structure

This paper contains 25 sections, 2 theorems, 34 equations, 7 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background and Related Work
  3. Problem Setup
  4. Related Work
  5. Identifiability of the Outcome Function
  6. Implementation of Flow IV
  7. Flow IV with Normalizing Flows
  8. Flow IV with Flow Matching
  9. Experiments
  10. Synthetic Data Experiments
  11. High-dimensional Outcome
  12. Real-world data: The Impact of Chinese Foreign Aid on Economic Development
  13. Discussion
  14. Technical Proofs
  15. Experiments
  16. ...and 10 more sections

Key Result

theorem 1

Let Assumption ass:triangular-monotonic and ass:strong-instrument be true. Let $g_{\bm{y}}: \mathbb{R}^m \times \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a function where $\bm{\tilde{u}_{\bm{y}}} \mapsto g_{\bm{y}}(\bm{a}, \bm{\tilde{u}_{\bm{y}}})$ is triangular monotonic following the same triangul where $\psi$ is an invertible function.

Figures (7)

  • Figure 1: Causal graph illustrating the structure of the type of DGP considered in this paper.
  • Figure 2: Comparison between Flow IV, Deep IV and GCFN in three different setups. Add. GCFN denotes the GCFN method with an additive decoder and Mult. GCFN with a multiplicative decoder. $\alpha$ controls the strength of confounding.
  • Figure 3: Comparison of counterfactual predictions for image data generated with Flow IV and a flow matching based associational model.
  • Figure 4: The outcome function learned with the Flow IV model in (b) and the expectation over $U_y$ in (a).
  • Figure 5: Finite sample convergence of Flow IV.
  • ...and 2 more figures

Theorems & Definitions (8)

  • theorem 1: Identifiability
  • Claim 1: Justification of Assumption \ref{['ass:triangular-monotonic']}
  • proof
  • lemma 1
  • proof
  • proof
  • Claim 2: Justification of the implication after Theorem \ref{['thrm:identifiabiliy']}
  • proof