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

arXiv: https://arxiv.org/abs/2608.15840 · HTML: https://mlsanity.com/paper/2608.15840/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 17 references that are in the map.
- First posted on arXiv 16 Aug 2026.

## Builds on (most cited first, 3 of 3)

- [High-dimensional learning of linear causal networks via inverse covariance estimation](https://mlsanity.com/paper/1311.3492/) (2013)
- [Large Dimensional Independent Component Analysis: Statistical Optimality and Computational Tractability](https://mlsanity.com/paper/2303.18156/) (2023)
- [Contrast-Free ICA and Causal Inference via Wasserstein Distances to the Gaussian](https://mlsanity.com/paper/2607.12832/) (2026)

## Abstract (arXiv)

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 nu measure each standardized disturbance's distance from Gaussianity, and let the disturbance scales lie in [underlineσ,overlineσ]. We prove the sharp local minimax law [ N2^star(β,nu,δ) asymp fraclog(1/δ) dβ2+β2nu2, qquad dβ= left[β2- left(1-fracunderlineσ2overlineσ2right)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.