确定因果方向需要多少个样本？双变量LiNGAM的精确极小极大边界
How Many Samples Are Needed to Determine Causal Direction? Sharp Minimax Bounds for Bivariate LiNGAM
AI总结：
该研究针对双变量LiNGAM，推导了确定因果方向所需样本量的精确极小极大边界，明确样本复杂度与边强度、非高斯性距离及尺度不确定性的关系，相关证明由GPT-5.6 Sol生成。
AI中文摘要：
我们研究确定两个线性相关变量之间因果方向所需的观测样本数量。经典LiNGAM理论表明，独立的非高斯扰动可识别因果方向，但未量化因果效应较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为每个标准化扰动与高斯性的距离，扰动尺度位于[underlineσ, overlineσ]区间内。我们证明了精确的局部极小极大定律：N₂^⋆(β,ν,δ) ≍ log(1/δ) / (d_β² + β²ν²)，其中d_β = [β² - (1 - underlineσ²/overlineσ²)]₊。现有理论仅确立了总体可识别性或假设两个方向间存在固定间隔，相比之下，我们将精确样本复杂度表征为边强度、非高斯性距离和尺度不确定性的联合函数，并明确了识别是源于非高斯依赖还是仅源于协方差。该证明由GPT-5.6 Sol在Codex的Ultra模式下，于两小时内独立生成，人类作者仅提供提示词，负责检查证明及修改润色手稿。
英文摘要：
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.
