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

• URL: http://arxiv.org/abs/2608.15840v1
• Date: Sun, 16 Aug 2026 16:20:19 GMT
• Title: How Many Samples Are Needed to Determine Causal Direction? Sharp Minimax Bounds for Bivariate LiNGAM
• Abstract summary: We study how many observations are needed to determine the causal direction between two linearly related variables.<n>We establish the sharp sample complexity as a joint function of edge strength, distance from Gaussianity, and scale uncertainty.
• Score: 7.7753818665096945
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• 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 $β$ bound the absolute structural coefficient from below, let $ν$ measure each standardized disturbance's distance from Gaussianity, and let the disturbance scales lie in $[\underlineσ,\overlineσ]$. We prove the sharp local minimax law \[ N_2^\star(β,ν,δ) \asymp \frac{\log(1/δ)} {d_β^2+β^2ν^2}, \qquad d_β= \left[β^2- \left(1-\frac{\underlineσ^2}{\overlineσ^2}\right)\right]_+. \] 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.

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