# GFCM: A Tail-Sensitive Mixed-Type Conditional Independence Test for Causal Discovery

arXiv: https://arxiv.org/abs/2608.15332 · HTML: https://mlsanity.com/paper/2608.15332/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 50 references that are in the map.
- First posted on arXiv 15 Aug 2026.

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

- [Causal Inference and Causal Explanation with Background Knowledge](https://mlsanity.com/paper/1302.4972/) (2013)
- [Causal Inference in the Presence of Latent Variables and Selection Bias](https://mlsanity.com/paper/1302.4983/) (2013)
- [Approximate Kernel-based Conditional Independence Tests for Fast Non-Parametric Causal Discovery](https://mlsanity.com/paper/1702.03877/) (2017)
- [Conditional independence testing based on a nearest-neighbor estimator of conditional mutual information](https://mlsanity.com/paper/1709.01447/) (2017)
- [On the Identifiability and Estimation of Causal Location-Scale Noise Models](https://mlsanity.com/paper/2210.09054/) (2022)
- [Cauchy combination test: a powerful test with analytic p-value calculation under arbitrary dependency structures](https://mlsanity.com/paper/1808.09011/) (2018)
- [Causal discovery in heavy-tailed models](https://mlsanity.com/paper/1908.05097/) (2019)
- [The Projected Covariance Measure for assumption-lean variable significance testing](https://mlsanity.com/paper/2211.02039/) (2022)
- [The Weighted Generalised Covariance Measure](https://mlsanity.com/paper/2111.04361/) (2021)
- [Conditional Independence Testing with Heteroskedastic Data and Applications to Causal Discovery](https://mlsanity.com/paper/2306.11498/) (2023)
- [Extremes of structural causal models](https://mlsanity.com/paper/2503.06536/) (2025)
- [Testing Conditional Independence via the Spectral Generalized Covariance Measure: Beyond Euclidean Data](https://mlsanity.com/paper/2511.15453/) (2025)
- [The Generalised Kernel Covariance Measure](https://mlsanity.com/paper/2604.03721/) (2026)
- [Causal Discovery in Multivariate Extremes via Tail Asymmetry](https://mlsanity.com/paper/2604.21620/) (2026)
- [Fourier Feature Methods for Nonlinear Causal Discovery: FFML Scoring, TRFF Scoring, and FFCI Testing in Mixed Data](https://mlsanity.com/paper/2605.05743/) (2026)
- [Fast Nonparametric Conditional Independence Testing via Two-Stage Regression](https://mlsanity.com/paper/2606.18011/) (2026)

## Abstract (arXiv)

Constraint-based causal discovery like PC and FCI depends on its conditional independence test. Partial correlation and the Generalised Covariance Measure (GCM) detect only the conditional covariance of residuals, so they miss dependence in the mean's nonlinear part, the scale, and the tails. Tests that detect more are biased inside PC, not scalable, only continuous, or not aimed at the tails. Our Generalised Feature Covariance Measure (GFCM) is valid, sensitive beyond covariance, robust inside PC, and applicable to mixed-type data. It runs the GCM template on a configurable set of residual features with conditional mean zero (centered moments and conditional quantile indicators), pooled in blocks and combined by the Cauchy rule, with a growing-knot spline nuisance at regression cost. We contribute (i) a centering result making the scale feature Neyman orthogonal, where the uncentered version is biased; (ii) the orientation asymmetry the mean-quantile construction creates inside PC, and its fix; (iii) a Phi-faithfulness theory under which PC with GFCM recovers the CPDAG of the set's detection class; and (iv) a benchmark of CI tests sensitive beyond covariance on synthetic data, semi-synthetic tail injections, and PC discovery on random DAGs. Under size-corrected power, GFCM recovers the scale and tail edges the covariance family misses and alone keeps power at the deep conditioning sets PC issues. It stays calibrated as n grows, whereas FFCI, the boosted GCM, and the partial copula test do not, and it handles mixed-type data directly. Inside PC at scale it attains the lowest skeleton SHD among tests that stay calibrated, while the others inflate false edges. Validity rests on an additive nuisance, and the tail advantage is shown on simulated and semi-synthetic data, as no fully real benchmark with both heavy tails and known structure exists.