定量给出确定两个变量因果方向所需的最少样本数。
How Many Samples Are Needed to Determine Causal Direction? Sharp Minimax Bounds for Bivariate LiNGAM
- 基于非高斯扰动和结构系数下界建模样本复杂度。
- 公式显示样本量与因果强度、非高斯性及尺度不确定性相关。
- 适用于弱因果或近高斯扰动场景，指导实际数据采集。
研究确定两个线性相关变量间因果方向所需观测样本数。经典LiNGAM理论表明独立非高斯扰动可识别因果方向，但未量化当因果效应微弱或扰动接近高斯分布时的难度。设β为结构系数绝对值的下界，ν衡量标准化扰动偏离高斯性的程度，扰动尺度位于[σ̲, σ̄]区间内。我们证明了精确的局部极小极大规律：N₂⋆(β, ν, δ) ≍ log(1/δ) / (d_β² + β²ν²)，其中d_β = [β² - (1 - σ̲²/σ̄²)]₊。此前理论仅说明总体可识别性或假设方向间固定分离。本文首次将样本复杂度建模为边强度、非高斯性距离和尺度不确定性的联合函数，并揭示识别机制是来自非高斯依赖还是协方差本身。证明由GPT-5.6 Sol在Codex Ultra模式下独立生成，人类作者仅负责提示输入并审核、修订与润色稿件。
原文摘要 · Abstract (English)
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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