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Conformal Inference of Individual Treatment Effects Using Conditional Density Estimates

Baozhen Wang, Xingye Qiao

TL;DR

This work tackles the challenge of constructing informative, valid prediction intervals for Individual Treatment Effects (ITE) under the potential outcomes framework with binary treatment. It proposes a two-stage conformal inference method that uses the conditional density $f(y|x)$ as the score, approximated efficiently via a reference distribution technique. The approach yields shorter prediction intervals while maintaining the nominal coverage, and it extends weighted conformal prediction with a two-stage framework to handle covariate shift in treatment groups. Empirical results on simulations and semi-synthetic benchmarks show that the proposed CD methods often outperform existing Weighted Conformal Prediction methods, offering practical gains for patient-specific decision-making in healthcare and policy contexts.

Abstract

In an era where diverse and complex data are increasingly accessible, the precise prediction of individual treatment effects (ITE) becomes crucial across fields such as healthcare, economics, and public policy. Current state-of-the-art approaches, while providing valid prediction intervals through Conformal Quantile Regression (CQR) and related techniques, often yield overly conservative prediction intervals. In this work, we introduce a conformal inference approach to ITE using the conditional density of the outcome given the covariates. We leverage the reference distribution technique to efficiently estimate the conditional densities as the score functions under a two-stage conformal ITE framework. We show that our prediction intervals are not only marginally valid but are narrower than existing methods. Experimental results further validate the usefulness of our method.

Conformal Inference of Individual Treatment Effects Using Conditional Density Estimates

TL;DR

This work tackles the challenge of constructing informative, valid prediction intervals for Individual Treatment Effects (ITE) under the potential outcomes framework with binary treatment. It proposes a two-stage conformal inference method that uses the conditional density as the score, approximated efficiently via a reference distribution technique. The approach yields shorter prediction intervals while maintaining the nominal coverage, and it extends weighted conformal prediction with a two-stage framework to handle covariate shift in treatment groups. Empirical results on simulations and semi-synthetic benchmarks show that the proposed CD methods often outperform existing Weighted Conformal Prediction methods, offering practical gains for patient-specific decision-making in healthcare and policy contexts.

Abstract

In an era where diverse and complex data are increasingly accessible, the precise prediction of individual treatment effects (ITE) becomes crucial across fields such as healthcare, economics, and public policy. Current state-of-the-art approaches, while providing valid prediction intervals through Conformal Quantile Regression (CQR) and related techniques, often yield overly conservative prediction intervals. In this work, we introduce a conformal inference approach to ITE using the conditional density of the outcome given the covariates. We leverage the reference distribution technique to efficiently estimate the conditional densities as the score functions under a two-stage conformal ITE framework. We show that our prediction intervals are not only marginally valid but are narrower than existing methods. Experimental results further validate the usefulness of our method.
Paper Structure (15 sections, 3 theorems, 21 equations, 1 figure, 1 table, 2 algorithms)

This paper contains 15 sections, 3 theorems, 21 equations, 1 figure, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Potential Outcome Framework and Objective Statement
  4. Conformal Prediction
  5. Weighted Conformal Prediction under Covariate Shift
  6. Methodology
  7. Conditional Density as the Score Function
  8. Estimate Conditional Density Using Reference Distribution Technique
  9. Weighted Conformal Inference Using Conditional Density Estimates
  10. Experimental Studies
  11. Experimental Setup
  12. Simulation Studies
  13. Semi-synthetic Datasets
  14. Results and Discussion
  15. Conclusions

Key Result

Theorem 1

Let $t_\alpha$ denote the $\alpha$ quantile of $f(Y|X=x)$. The solution that optimizes eq:obj is given by $C_{\alpha}^* = \left\{(x,y) : f(y|x) \geq t_{\alpha} \right\}$. And the optimal predictor can be written as

Figures (1)

  • Figure 1: Performance of all baselines in four simulation scenarios described in Section \ref{['subsec:simu']}. The red vertical lines correspond to target coverage, and the blue vertical lines correspond to the optimal interval width. Here CD stands for Algorithm \ref{['alg1']}, WCP stands for weighted conformal prediction, and CM stands for conformal meta-learners.

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Lemma 1: lei2021conformal