Generalised Kernel Covariance Measure (GKCM) Test

GKCM (Bergen, Sejdinovic & Didelez, CLeaR 2025) is a kernelised generalisation of the Generalised Covariance Measure. Where GCM tests whether the scalar residuals \(X - \hat{E}[X\mid Z]\) and \(Y - \hat{E}[Y\mid Z]\) are uncorrelated, GKCM maps the residuals into a reproducing-kernel Hilbert space and tests whether their kernel cross-covariance operator is zero, so it detects dependence in the full conditional distribution rather than only in the conditional mean (Bergen et al., 2025).

Intuition. Regress \(X\) and \(Y\) on \(Z\) with a flexible nuisance, embed the residuals with characteristic kernels, and test the Hilbert–Schmidt norm of their cross-covariance. Under a characteristic kernel the statistic is zero exactly when the residual embeddings are independent, widening GCM’s mean-zero detection class toward full conditional independence (Bergen et al., 2025).

Assumptions

  • Continuous data. GKCM is formulated for continuous variables (Bergen et al., 2025).

  • R + comets available. This wrapper dispatches to the R comets::kgcm implementation and requires rpy2 and the R comets package (install.packages("comets")).

  • Nuisance rate. Like GCM, validity rests on the nuisance regressions estimating the conditional means at a sufficient rate; the regression method is the R reg_YonZ/reg_XonZ argument (default "rf"), set through the reg keyword.

  • Quadratic cost. The kernel statistic is \(O(n^2)\) in the sample size, so it does not scale to large \(n\) the way the random-feature relaxations (rcit, rcot) do.

Code Example

import numpy as np
from citests.tests import GKCM

rng = np.random.default_rng(0)
n = 300
z = rng.normal(size=n)
x = z + 0.5 * rng.normal(size=n)
y = z + 0.5 * rng.normal(size=n)          # x _||_ y | z
data = np.column_stack([x, y, z])

p = GKCM(data)(0, 1, [2])                 # test X _||_ Y | Z
print(p)

# select the nuisance regression (R comets method name), default "rf":
p_lm = GKCM(data, reg="lasso")(0, 1, [2])

References

  • Bergen, L., Sejdinovic, D., & Didelez, V. (2025). The Generalised Kernel Covariance Measure. Proceedings of the Conference on Causal Learning and Reasoning (CLeaR).

  • Shah, R. D., & Peters, J. (2020). The Hardness of Conditional Independence Testing and the Generalised Covariance Measure. Annals of Statistics.