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Extended Fiducial Inference for Individual Treatment Effects via Deep Neural Networks

Sehwan Kim, Faming Liang

TL;DR

The paper addresses uncertainty quantification in individual treatment effect (ITE) estimation by extending Extended Fiducial Inference (EFI) to large neural models. It introduces a Double-NN framework that models the treatment and control functions with DNNs while learning an inverse mapping via a second network, enabling valid inference as model size grows at a rate $O(n^{ta})$ with $0\leq \u03b7<1$. The authors prove consistency results for the learned inverse mapping under large-model regimes and demonstrate, through simulations and real data analyses, that Double-NN yields tighter, near-nominal predictive ITE intervals and improved CATE inference compared to conformal quantile regression. The approach provides a principled, scalable method for uncertainty quantification in deep learning-based causal inference, with practical benefits for policy evaluation and personalized treatment decisions in high-dimensional settings.

Abstract

Individual treatment effect estimation has gained significant attention in recent data science literature. This work introduces the Double Neural Network (Double-NN) method to address this problem within the framework of extended fiducial inference (EFI). In the proposed method, deep neural networks are used to model the treatment and control effect functions, while an additional neural network is employed to estimate their parameters. The universal approximation capability of deep neural networks ensures the broad applicability of this method. Numerical results highlight the superior performance of the proposed Double-NN method compared to the conformal quantile regression (CQR) method in individual treatment effect estimation. From the perspective of statistical inference, this work advances the theory and methodology for statistical inference of large models. Specifically, it is theoretically proven that the proposed method permits the model size to increase with the sample size $n$ at a rate of $O(n^ζ)$ for some $0 \leq ζ<1$, while still maintaining proper quantification of uncertainty in the model parameters. This result marks a significant improvement compared to the range $0\leq ζ< \frac{1}{2}$ required by the classical central limit theorem. Furthermore, this work provides a rigorous framework for quantifying the uncertainty of deep neural networks under the neural scaling law, representing a substantial contribution to the statistical understanding of large-scale neural network models.

Extended Fiducial Inference for Individual Treatment Effects via Deep Neural Networks

TL;DR

The paper addresses uncertainty quantification in individual treatment effect (ITE) estimation by extending Extended Fiducial Inference (EFI) to large neural models. It introduces a Double-NN framework that models the treatment and control functions with DNNs while learning an inverse mapping via a second network, enabling valid inference as model size grows at a rate with . The authors prove consistency results for the learned inverse mapping under large-model regimes and demonstrate, through simulations and real data analyses, that Double-NN yields tighter, near-nominal predictive ITE intervals and improved CATE inference compared to conformal quantile regression. The approach provides a principled, scalable method for uncertainty quantification in deep learning-based causal inference, with practical benefits for policy evaluation and personalized treatment decisions in high-dimensional settings.

Abstract

Individual treatment effect estimation has gained significant attention in recent data science literature. This work introduces the Double Neural Network (Double-NN) method to address this problem within the framework of extended fiducial inference (EFI). In the proposed method, deep neural networks are used to model the treatment and control effect functions, while an additional neural network is employed to estimate their parameters. The universal approximation capability of deep neural networks ensures the broad applicability of this method. Numerical results highlight the superior performance of the proposed Double-NN method compared to the conformal quantile regression (CQR) method in individual treatment effect estimation. From the perspective of statistical inference, this work advances the theory and methodology for statistical inference of large models. Specifically, it is theoretically proven that the proposed method permits the model size to increase with the sample size at a rate of for some , while still maintaining proper quantification of uncertainty in the model parameters. This result marks a significant improvement compared to the range required by the classical central limit theorem. Furthermore, this work provides a rigorous framework for quantifying the uncertainty of deep neural networks under the neural scaling law, representing a substantial contribution to the statistical understanding of large-scale neural network models.
Paper Structure (29 sections, 7 theorems, 67 equations, 7 figures, 4 tables, 1 algorithm)

This paper contains 29 sections, 7 theorems, 67 equations, 7 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. A Brief Review of the EFI Method
  3. EFI for Large Models
  4. Consistency of Inverse Mapping Learned in EFI for Large Models
  5. Double-NN Method
  6. An Illustrative Example for EFI
  7. Causal Inference for Individual Treatment Effects
  8. ITE prediction intervals
  9. Simulation Study
  10. Example 1
  11. Example 2
  12. Precision in Estimation of Heterogeneous Effects
  13. Real Data Analysis
  14. Lalonde
  15. NLSM
  16. ...and 14 more sections

Key Result

Theorem 3.1

Suppose Assumptions ass1-ass6 hold (see the supplement), $\epsilon$ is sufficiently small, and where $d_l$ denotes the width of layer $l$, $d_H=dim({\boldsymbol \theta})$, and $H$ denotes the depth of the DNN in the EFI network. Then $G^*({\boldsymbol Y}_n,{\boldsymbol X}_n,{\boldsymbol Z}_n)= \frac{1}{n} \sum_{i=1}^n \hat{g}(y_i,{\boldsymbol x}_i,z_i,{\boldsymbol w}_n^*)$ constitutes a consi

Figures (7)

  • Figure 1: Illustration of the EFI network LiangKS2024EFI, where the orange nodes and orange links form a DNN (parameterized by the weights ${\boldsymbol w}_n$, with the subscript $n$ indicating its dependence on the training sample size $n$), the green node represents latent variable to impute, and the black lines represent deterministic functions.
  • Figure 2: Demonstration of the Double-NN method for a dataset simulated from (\ref{['dataGex1']}): (left) scatter plot of $\hat{{\boldsymbol z}}_i$ ($y$-axis) versus ${\boldsymbol z}_i$ ($x$-axis); (middle) Q-Q plot of $\hat{{\boldsymbol z}}_i$ and ${\boldsymbol z}_i$; (right) scatter plot of $\tau({\boldsymbol x}_i)$ ($y$-axis) versus $\hat{\tau}({\boldsymbol x}_i)$ ($x$-axis).
  • Figure 3: Demonstration of the Double-NN method for a dataset simulated from (\ref{['dataGex2']}): (left) scatter plot of $\hat{{\boldsymbol z}}_i$ ($y$-axis) versus ${\boldsymbol z}_i$ ($x$-axis); (middle-left) Q-Q plot of $\hat{{\boldsymbol z}}_i$ and ${\boldsymbol z}_i$; (middle-right) scatter plot of $c({\boldsymbol x}_i)$ ($y$-axis) versus $\hat{c}({\boldsymbol x}_i)$ ($x$-axis); (right) scatter plot of $\tau({\boldsymbol x}_i)$ ($y$-axis) versus $\hat{\tau}({\boldsymbol x}_i)$ ($x$-axis).
  • Figure 4: Comparison of prediction intervals resulting from Double-NN (labeled as EFI) and CQR (labeled as conformal) for the subjects in the test set of Lalonde.
  • Figure 5: Comparison of the average length of intervals obtained by the Double-NN (labeled as EFI) and CQR (labeled as conformal) for the NLSM data.
  • ...and 2 more figures

Theorems & Definitions (10)

  • Theorem 3.1
  • Remark 1
  • Lemma S1
  • Remark S1
  • Lemma S2
  • Lemma S3
  • Lemma S4
  • Theorem S1
  • Theorem S2
  • Remark S2