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Estimating Aleatoric Uncertainty in the Causal Treatment Effect

Liyuan Xu, Bijan Mazaheri

TL;DR

The paper introduces the variance of the treatment effect $\mathrm{VTE}$ and the conditional variance $\mathrm{CVTE}(v)$ as principled measures of aleatoric uncertainty in causal treatment effects. It establishes identifiability from observational data under ignorability plus a discompositional assumption that $\mathbb{E}[Y^{(0)}Y^{(1)}|X]=\mathbb{E}[Y^{(0)}|X]\mathbb{E}[Y^{(1)}|X]$, and derives kernel-based nonparametric estimators with convergence guarantees for $f_a$ and $g_a$ used in the VTE/CVTE expressions. The estimators leverage kernel ridge regression and conditional mean embeddings to produce consistent estimates of $\mathrm{VTE}$ and $\mathrm{CVTE}$, respectively, and are validated on synthetic and semi-simulated data where they outperform naive baselines and demonstrate competitive performance. Overall, the work provides a rigorous, scalable framework for quantifying and decomposing intrinsic uncertainty in causal treatment effects, with potential implications for risk assessment in personalized interventions.

Abstract

Previous work on causal inference has primarily focused on averages and conditional averages of treatment effects, with significantly less attention on variability and uncertainty in individual treatment responses. In this paper, we introduce the variance of the treatment effect (VTE) and conditional variance of treatment effect (CVTE) as the natural measure of aleatoric uncertainty inherent in treatment responses, and we demonstrate that these quantities are identifiable from observed data under mild assumptions, even in the presence of unobserved confounders. We further propose nonparametric kernel-based estimators for VTE and CVTE, and our theoretical analysis establishes their convergence. We also test the performance of our method through extensive empirical experiments on both synthetic and semi-simulated datasets, where it demonstrates superior or comparable performance to naive baselines.

Estimating Aleatoric Uncertainty in the Causal Treatment Effect

TL;DR

The paper introduces the variance of the treatment effect and the conditional variance as principled measures of aleatoric uncertainty in causal treatment effects. It establishes identifiability from observational data under ignorability plus a discompositional assumption that , and derives kernel-based nonparametric estimators with convergence guarantees for and used in the VTE/CVTE expressions. The estimators leverage kernel ridge regression and conditional mean embeddings to produce consistent estimates of and , respectively, and are validated on synthetic and semi-simulated data where they outperform naive baselines and demonstrate competitive performance. Overall, the work provides a rigorous, scalable framework for quantifying and decomposing intrinsic uncertainty in causal treatment effects, with potential implications for risk assessment in personalized interventions.

Abstract

Previous work on causal inference has primarily focused on averages and conditional averages of treatment effects, with significantly less attention on variability and uncertainty in individual treatment responses. In this paper, we introduce the variance of the treatment effect (VTE) and conditional variance of treatment effect (CVTE) as the natural measure of aleatoric uncertainty inherent in treatment responses, and we demonstrate that these quantities are identifiable from observed data under mild assumptions, even in the presence of unobserved confounders. We further propose nonparametric kernel-based estimators for VTE and CVTE, and our theoretical analysis establishes their convergence. We also test the performance of our method through extensive empirical experiments on both synthetic and semi-simulated datasets, where it demonstrates superior or comparable performance to naive baselines.
Paper Structure (34 sections, 11 theorems, 52 equations, 4 figures, 3 tables)

This paper contains 34 sections, 11 theorems, 52 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Works
  3. Heterogeneous treatment effects
  4. Counterfactual Distributions
  5. Moments of the Causal Effect
  6. Variance of ATE Estimators
  7. Kernel Mean Embedding in Causal Inference
  8. Problem Setting
  9. VTE Definition
  10. Motivating Example
  11. Extension to Conditional VTE
  12. Identification
  13. Estimation
  14. Preliminary
  15. Reproducing Kernel Hilbert Space
  16. ...and 19 more sections

Key Result

Theorem 4.2

There exists two data generation processes $P(X,A,Y^{(0)}, Y^{(1)})$ satisfying assum:ignorability that have different VTE but the same observational distribution $P(X,A,Y)$.

Figures (4)

  • Figure 1: Histogram of treatment effect $Y^{(1)} - Y^{(0)}$
  • Figure 2: A causal graph satisfying the constraints of our setting. The dotted circles indicate that the potential outcomes are not observed.
  • Figure 3: Box plot for VTE prediction. The red line shows the true VTE.
  • Figure 4: Box plot for CVTE prediction. The red line shows the true CVTE.

Theorems & Definitions (16)

  • Definition 3.1
  • Definition 3.2
  • Theorem 4.2
  • Theorem 4.4
  • Theorem 4.5
  • Remark 4.6
  • Proposition 5.1: Fischer2020Sobolev
  • Theorem 5.2
  • Lemma 5.3
  • Proposition 5.4: singh2019kernel
  • ...and 6 more