# Sample Complexity of Nonparametric Closeness Testing for Continuous Distributions and Its Application to Causal Discovery with Hidden Confounding

arXiv: https://arxiv.org/abs/2503.07475 · HTML: https://mlsanity.com/paper/2503.07475/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 58 references that are in the map.
- First posted on arXiv 10 Mar 2025.

## Builds on (most cited first, 19 of 19)

- [A Tutorial on Kernel Density Estimation and Recent Advances](https://mlsanity.com/paper/1704.03924/) (2017)
- [Characterizing and Learning Equivalence Classes of Causal DAGs under Interventions](https://mlsanity.com/paper/1802.06310/) (2018)
- [Learning Causal Graphs with Small Interventions](https://mlsanity.com/paper/1511.00041/) (2015)
- [Cost-Optimal Learning of Causal Graphs](https://mlsanity.com/paper/1703.02645/) (2017)
- [Causal Discovery from Temporal Data: An Overview and New Perspectives](https://mlsanity.com/paper/2303.10112/) (2023)
- [Active Structure Learning of Causal DAGs via Directed Clique Tree](https://mlsanity.com/paper/2011.00641/) (2020)
- [The Influence Function of Semiparametric Estimators](https://mlsanity.com/paper/1508.01378/) (2015)
- [Testing Conditional Independence of Discrete Distributions](https://mlsanity.com/paper/1711.11560/) (2017)
- [A Bayesian Approach to Learning Bayesian Networks with Local Structure](https://mlsanity.com/paper/1302.1528/) (2013)
- [Optimal Testing of Discrete Distributions with High Probability](https://mlsanity.com/paper/2009.06540/) (2020)
- [Learning Bayesian Networks in the Presence of Structural Side Information](https://mlsanity.com/paper/2112.10884/) (2021)
- [Optimal Algorithms and Lower Bounds for Testing Closeness of Structured Distributions](https://mlsanity.com/paper/1508.05538/) (2015)
- [Direct Estimation of Information Divergence Using Nearest Neighbor Ratios](https://mlsanity.com/paper/1702.05222/) (2017)
- [Minimax Optimal Estimation of KL Divergence for Continuous Distributions](https://mlsanity.com/paper/2002.11599/) (2020)
- [On the Sample Complexity of Causal Discovery and the Value of Domain Expertise](https://mlsanity.com/paper/2102.03274/) (2021)
- [Testing Closeness of Multivariate Distributions via Ramsey Theory](https://mlsanity.com/paper/2311.13154/) (2023)
- [Exponential Concentration of a Density Functional Estimator](https://mlsanity.com/paper/1603.08584/) (2016)
- [On sample complexity of conditional independence testing with Von Mises estimator with application to causal discovery](https://mlsanity.com/paper/2310.13553/) (2023)
- [Gaussian Mean Testing Made Simple](https://mlsanity.com/paper/2210.13706/) (2022)

## Abstract (arXiv)

We study the problem of closeness testing for continuous distributions and its implications for causal discovery. Specifically, we analyze the sample complexity of distinguishing whether two multidimensional continuous distributions are identical or differ by at least ε in terms of Kullback-Leibler (KL) divergence under non-parametric assumptions. To this end, we propose an estimator of KL divergence which is based on the von Mises expansion. Our closeness test attains optimal parametric rates under smoothness assumptions. Equipped with this test, which serves as a building block of our causal discovery algorithm to identify the causal structure between two multidimensional random variables, we establish sample complexity guarantees for our causal discovery method. To the best of our knowledge, this work is the first work that provides sample complexity guarantees for distinguishing cause and effect in multidimensional non-linear models with non-Gaussian continuous variables in the presence of unobserved confounding.