How Many Samples Are Needed to Determine Causal Direction? Sharp Minimax Bounds for Bivariate LiNGAM

Jikai Jin

Papers from arXiv.org

Abstract: We study how many observations are needed to determine the causal direction between two linearly related variables. Classical LiNGAM theory shows that independent non-Gaussian disturbances identify the direction, but does not quantify the difficulty when the causal effect is weak or the disturbances are nearly Gaussian. Let $\beta$ bound the absolute structural coefficient from below, let $\nu$ measure each standardized disturbance's distance from Gaussianity, and let the disturbance scales lie in $[\underline\sigma,\overline\sigma]$. We prove the sharp local minimax law \[ N_2^\star(\beta,\nu,\delta) \asymp \frac{\log(1/\delta)} {d_\beta^2+\beta^2\nu^2}, \qquad d_\beta= \left[\beta^2- \left(1-\frac{\underline\sigma^2}{\overline\sigma^2}\right)\right]_+. \] Previous theory established population identifiability or assumed a fixed separation between the two directions. By contrast, we establish the sharp sample complexity as a joint function of edge strength, distance from Gaussianity, and scale uncertainty, and characterize when identification comes from non-Gaussian dependence or from covariance alone. The proof was independently generated with GPT-5.6 Sol in Codex's Ultra mode during a two-hour session. The human author supplied the prompt and was responsible only forchecking the proof and revising and polishing the manuscript.

Date: 2026-08
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