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

2026-08-16

AI Summary

This study addresses the sample complexity of causal identification in bivariate linear non-Gaussian models by integrating non-Gaussian independent component analysis with minimax theory. It establishes, for the first time, sharp local minimax bounds that depend on edge strength, non-Gaussianity, and scale uncertainty. The derived exact sample size formula elucidates identification mechanisms under weak-effect or near-Gaussian regimes and characterizes critical conditions distinguishing non-Gaussianity-dominated from covariance-dominated identification. By providing rigorous theoretical support and quantitative criteria, this work significantly advances the understanding of statistical limits in determining causal directionality within linear non-Gaussian frameworks.

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.

Problem

• Causal Direction
• Sample Complexity
• LiNGAM
• Minimax Bounds
• Non-Gaussianity

Innovation

• Sharp Minimax Bounds
• Sample Complexity
• Bivariate LiNGAM
• Non-Gaussianity
• AI-generated Proof