# Conditional Independence Testing in Hilbert Spaces with Applications to Functional Data Analysis

arXiv: https://arxiv.org/abs/2101.07108 · HTML: https://mlsanity.com/paper/2101.07108/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Jan 2021.

## Built on by (most cited first, 12 of 12)

- [Higher Order Kernel Mean Embeddings to Capture Filtrations of Stochastic Processes](https://mlsanity.com/paper/2109.03582/) (2021)
- [Signature Kernel Conditional Independence Tests in Causal Discovery for Stochastic Processes](https://mlsanity.com/paper/2402.18477/) (2024)
- [High-dimensional Functional Graphical Model Structure Learning via Neighborhood Selection Approach](https://mlsanity.com/paper/2105.02487/) (2021)
- [Nonparametric Conditional Local Independence Testing](https://mlsanity.com/paper/2203.13559/) (2022)
- [Model-based causal feature selection for general response types](https://mlsanity.com/paper/2309.12833/) (2023)
- [Deep Nonparametric Conditional Independence Tests for Images](https://mlsanity.com/paper/2411.06140/) (2024)
- [An Asymmetric Independence Model for Causal Discovery on Path Spaces](https://mlsanity.com/paper/2503.09859/) (2025)
- [Cluster-Dags as Powerful Background Knowledge For Causal Discovery](https://mlsanity.com/paper/2512.10032/) (2025)
- [The Generalised Kernel Covariance Measure](https://mlsanity.com/paper/2604.03721/) (2026)
- [FACT: High-Dimensional Random Forests Inference](https://mlsanity.com/paper/2207.01678/) (2022)
- [Your Assumed DAG is Wrong and Here's How To Deal With It](https://mlsanity.com/paper/2502.17030/) (2025)
- [Marginal Coordinate Test for Fr'echet Regression with Random Objects](https://mlsanity.com/paper/2608.30644/) (2026)

## Abstract (arXiv)

We study the problem of testing the null hypothesis that X and Y are conditionally independent given Z, where each of X, Y and Z may be functional random variables. This generalises testing the significance of X in a regression model of scalar response Y on functional regressors X and Z. We show however that even in the idealised setting where additionally (X, Y, Z) has a Gaussian distribution, the power of any test cannot exceed its size. Further modelling assumptions are needed and we argue that a convenient way of specifying these is based on choosing methods for regressing each of X and Y on Z. We propose a test statistic involving inner products of the resulting residuals that is simple to compute and calibrate: type I error is controlled uniformly when the in-sample prediction errors are sufficiently small. We show this requirement is met by ridge regression in functional linear model settings without requiring any eigen-spacing conditions or lower bounds on the eigenvalues of the covariance of the functional regressor. We apply our test in constructing confidence intervals for truncation points in truncated functional linear models and testing for edges in a functional graphical model for EEG data.