# Extended fiducial inference for individual treatment effects via deep neural networks

## TL;DR

This paper introduces a new method using deep neural networks to estimate individual treatment effects with improved statistical inference and uncertainty quantification.

## Contribution

The paper introduces the Double-NN method under extended fiducial inference, allowing model size to scale with sample size while maintaining uncertainty quantification.

## Key findings

- The Double-NN method outperforms conformal quantile regression in individual treatment effect estimation.
- The proposed method allows model size to grow at a rate of O(n^ζ) for 0 ≤ ζ < 1 while maintaining uncertainty quantification.
- A rigorous framework is provided for uncertainty quantification in deep neural networks under the neural scaling law.

## 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 \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$O(n^{\zeta })$$\end{document}O(nζ) for some \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$0 \le \zeta <1$$\end{document}0≤ζ<1, while still maintaining proper quantification of uncertainty in the model parameters. This result marks a significant improvement compared to the range \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$0\le \zeta < \frac{1}{2}$$\end{document}0≤ζ<12 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.

The online version contains supplementary material available at 10.1007/s11222-025-10624-8.