The Generalised Kernel Covariance Measure

Testing conditional independence is fundamental yet intrinsically difficult: without additional assumptions, Type I error control is impossible in general. The "Model-X'' paradigm addresses this difficulty by assuming exact knowledge of a relevant conditional distribution. While small deviations from this assumption can sometimes be tolerated in classical one-shot testing, existing sequential conditional independence tests typically require the Model-X conditional to be known exactly, making them fragile when it must instead be estimated. We propose a new approach that is substantially more robust to such estimation error. Our method applies testing-by-betting to an adaptively optimized Kernel Conditional Independence statistic, together with a normalization scheme and a truncate-and-shift calibration strategy. These modifications greatly reduce Type I error inflation while preserving high power across high-dimensional synthetic benchmarks and real-world fairness tasks, outperforming existing sequential Model-X approaches. Code is available at https://github.com/he-zh/SKCI.

Comparing conditional distributions is a fundamental challenge in statistics and machine learning, with applications across a wide range of domains. While proposed methods for measuring discrepancies using kernel embeddings of distributions in a reproducing kernel Hilbert space (RKHS) provide powerful non-parametric techniques, the existing literature remains fragmented and lacks a unified theoretical treatment. This paper addresses this gap by establishing a coherent framework for studying kernel-based methods to measure divergence between conditional distributions through what we refer to as conditional maximum mean discrepancy (CMMD). The CMMD consists of a family of metrics which we call levels, with three special cases each using a different type of RKHS embedding: CMMD$_0$ (conditional mean operators), CMMD$_1$ (conditional mean embeddings), and CMMD$_2$ (joint mean embeddings). We additionally introduce a general level $s$ CMMD, clarifying the required assumptions, and establishing mathematical connections between the levels through the lens of operator-based smoothing. In addition to reviewing previously proposed estimators, we introduce a novel doubly robust estimator for the CMMD that maintains consistency provided the product of errors of the underlying models reduces sufficiently fast. We provide numerical experiments demonstrating that the CMMD effectively captures complex conditional dependencies for statistical testing.

We introduce kernel integrated $R^2$, a new measure of statistical dependence that combines the local normalization principle of the recently introduced integrated $R^2$ with the flexibility of reproducing kernel Hilbert spaces (RKHSs). The proposed measure extends integrated $R^2$ from scalar responses to responses taking values on general spaces equipped with a characteristic kernel, allowing to measure dependence of multivariate, functional, and structured data, while remaining sensitive to tail behaviour and oscillatory dependence structures. We establish that (i) this new measure takes values in $[0,1]$, (ii) equals zero if and only if independence holds, and (iii) equals one if and only if the response is almost surely a measurable function of the covariates. Two estimators are proposed: a graph-based method using $K$-nearest neighbours and an RKHS-based method built on conditional mean embeddings. We prove consistency and derive convergence rates for the graph-based estimator, showing its adaptation to intrinsic dimensionality. Numerical experiments on simulated data and a real data experiment in the context of dependency testing for media annotations demonstrate competitive power against state-of-the-art dependence measures, particularly in settings involving non-linear and structured relationships.