# Partial identification of kernel based two sample tests with mismeasured data

arXiv: https://arxiv.org/abs/2308.03570 · HTML: https://mlsanity.com/paper/2308.03570/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 35 references that are in the map.
- First posted on arXiv 7 Aug 2023.

## Builds on (most cited first, 16 of 16)

- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [Stabilizing Off-Policy Q-Learning via Bootstrapping Error Reduction](https://mlsanity.com/paper/1906.00949/) (2019)
- [Learning Adversarially Fair and Transferable Representations](https://mlsanity.com/paper/1802.06309/) (2018)
- [MMD GAN: Towards Deeper Understanding of Moment Matching Network](https://mlsanity.com/paper/1705.08584/) (2017)
- [The Variational Fair Autoencoder](https://mlsanity.com/paper/1511.00830/) (2015)
- [Assessing Algorithmic Fairness with Unobserved Protected Class Using Data Combination](https://mlsanity.com/paper/1906.00285/) (2019)
- [Distributionally Robust Optimization and Generalization in Kernel Methods](https://mlsanity.com/paper/1905.10943/) (2019)
- [Interpretable Distribution Features with Maximum Testing Power](https://mlsanity.com/paper/1605.06796/) (2016)
- [Distributionally Robust Bayesian Optimization](https://mlsanity.com/paper/2002.09038/) (2020)
- [Causally motivated Shortcut Removal Using Auxiliary Labels](https://mlsanity.com/paper/2105.06422/) (2021)
- [Toward a better trade-off between performance and fairness with kernel-based distribution matching](https://mlsanity.com/paper/1910.11779/) (2019)
- [Efficient Aggregated Kernel Tests using Incomplete U-statistics](https://mlsanity.com/paper/2206.09194/) (2022)
- [Compress Then Test: Powerful Kernel Testing in Near-linear Time](https://mlsanity.com/paper/2301.05974/) (2023)
- [Fairness and robustness in anti-causal prediction](https://mlsanity.com/paper/2209.09423/) (2022)
- [Learning Invariant Representations with Missing Data](https://mlsanity.com/paper/2112.00881/) (2021)
- [A Reproducing Kernel Hilbert Space log-rank test for the two-sample problem](https://mlsanity.com/paper/1904.05187/) (2019)

## Abstract (arXiv)

Nonparametric two-sample tests such as the Maximum Mean Discrepancy (MMD) are often used to detect differences between two distributions in machine learning applications. However, the majority of existing literature assumes that error-free samples from the two distributions of interest are available.We relax this assumption and study the estimation of the MMD under ε-contamination, where a possibly non-random ε proportion of one distribution is erroneously grouped with the other. We show that under ε-contamination, the typical estimate of the MMD is unreliable. Instead, we study partial identification of the MMD, and characterize sharp upper and lower bounds that contain the true, unknown MMD. We propose a method to estimate these bounds, and show that it gives estimates that converge to the sharpest possible bounds on the MMD as sample size increases, with a convergence rate that is faster than alternative approaches. Using three datasets, we empirically validate that our approach is superior to the alternatives: it gives tight bounds with a low false coverage rate.