# Distance and Kernel-Based Measures for Global and Local Two-Sample Conditional Distribution Testing

arXiv: https://arxiv.org/abs/2210.08149 · HTML: https://mlsanity.com/paper/2210.08149/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 79 references that are in the map.
- First posted on arXiv 15 Oct 2022.

## Builds on (most cited first, 15 of 15)

- [Conformal Prediction Under Covariate Shift](https://mlsanity.com/paper/1904.06019/) (2019)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [An introduction to domain adaptation and transfer learning](https://mlsanity.com/paper/1812.11806/) (2018)
- [A Measure-Theoretic Approach to Kernel Conditional Mean Embeddings](https://mlsanity.com/paper/2002.03689/) (2020)
- [Optimally tackling covariate shift in RKHS-based nonparametric regression](https://mlsanity.com/paper/2205.02986/) (2022)
- [On the Optimality of Kernel-Embedding Based Goodness-of-Fit Tests](https://mlsanity.com/paper/1709.08148/) (2017)
- [A Two-Sample Conditional Distribution Test Using Conformal Prediction and Weighted Rank Sum](https://mlsanity.com/paper/2010.07147/) (2020)
- [Global and Local Two-Sample Tests via Regression](https://mlsanity.com/paper/1812.08927/) (2018)
- [Augmented Transfer Regression Learning with Semi-non-parametric Nuisance Models](https://mlsanity.com/paper/2010.02521/) (2020)
- [Distance Metrics for Measuring Joint Dependence with Application to Causal Inference](https://mlsanity.com/paper/1711.09179/) (2017)
- [Nonparametric Tests for Treatment Effect Heterogeneity with Duration Outcomes](https://mlsanity.com/paper/1612.02090/) (2016)
- [A New Framework for Distance and Kernel-based Metrics in High Dimensions](https://mlsanity.com/paper/1909.13469/) (2019)
- [Distance-based and RKHS-based Dependence Metrics in High Dimension](https://mlsanity.com/paper/1902.03291/) (2019)
- [Kernel Two-Sample Tests in High Dimension: Interplay Between Moment Discrepancy and Dimension-and-Sample Orders](https://mlsanity.com/paper/2201.00073/) (2021)
- [Testing mutual independence in high dimension via distance covariance](https://mlsanity.com/paper/1609.09380/) (2016)

## Built on by (most cited first, 7 of 7)

- [A Kernel-Based Conditional Two-Sample Test Using Nearest Neighbors (with Applications to Calibration, Regression Curves, and Simulation-Based Inference)](https://mlsanity.com/paper/2407.16550/) (2024)
- [General Frameworks for Conditional Two-Sample Testing](https://mlsanity.com/paper/2410.16636/) (2024)
- [Measuring Differences between Conditional Distributions using Kernel Embeddings](https://mlsanity.com/paper/2605.02260/) (2026)
- [Kernel conditional tests from learning-theoretic bounds](https://mlsanity.com/paper/2506.03898/) (2025)
- [Machine-Learning-Assisted Comparison of Regression Functions](https://mlsanity.com/paper/2510.24714/) (2025)
- [Existence-Field Diffusion Model for Spatial Point Processes with Variable Cardinality](https://mlsanity.com/paper/2607.26428/) (2026)
- [Testing Conditional Distribution Equality Using Generative Models](https://mlsanity.com/paper/2610.20447/) (2026)

## Abstract (arXiv)

Testing the equality of two conditional distributions is crucial in various modern applications, including transfer learning and causal inference. Despite its importance, this fundamental problem has received surprisingly little attention in the literature, with existing works focusing exclusively on global two-sample conditional distribution testing. Based on distance and kernel methods, this paper presents the first unified framework for both global and local two-sample conditional distribution testing. To this end, we introduce distance and kernel-based measures that characterize the homogeneity of two conditional distributions. Drawing from the concept of conditional U-statistics, we propose consistent estimators for these measures. Theoretically, we derive the convergence rates and the asymptotic distributions of the estimators under both the null and alternative hypotheses. Utilizing these measures, along with a local bootstrap approach, we develop global and local tests that can detect discrepancies between two conditional distributions at global and local levels, respectively. Our tests demonstrate reliable performance through simulations and real data analysis.