Additive Noise Models (ANMs) are a common model class for causal discovery from observational data and are often used to generate synthetic data for causal discovery benchmarking. Specifying an ANM requires choosing all parameters, including those not fixed by explicit assumptions. Reisach et al. (2021) show that sorting variables by increasing variance often yields an ordering close to a causal order and introduce var-sortability to quantify this alignment. Since increasing variances may be unrealistic and are scale-dependent, ANM data are often standardized in benchmarks. We show that synthetic ANM data are characterized by another pattern that is scale-invariant: the explainable fraction of a variable's variance, as captured by the coefficient of determination $R^2$, tends to increase along the causal order. The result is high $R^2$-sortability, meaning that sorting the variables by increasing $R^2$ yields an ordering close to a causal order. We propose an efficient baseline algorithm termed $R^2$-SortnRegress that exploits high $R^2$-sortability and that can match and exceed the performance of established causal discovery algorithms. We show analytically that sufficiently high edge weights lead to a relative decrease of the noise contributions along causal chains, resulting in increasingly deterministic relationships and high $R^2$. We characterize $R^2$-sortability for different simulation parameters and find high values in common settings. Our findings reveal high $R^2$-sortability as an assumption about the data generating process relevant to causal discovery and implicit in many ANM sampling schemes. It should be made explicit, as its prevalence in real-world data is unknown. For causal discovery benchmarking, we implement $R^2$-sortability, the $R^2$-SortnRegress algorithm, and ANM simulation procedures in our library CausalDisco at https://causaldisco.github.io/CausalDisco/.

翻译：加性噪声模型（ANMs）是从观测数据中进行因果发现的常见模型类别，常被用于生成因果发现基准测试的合成数据。指定一个ANM需要选择所有参数，包括那些未被明确假设固定的参数。Reisach等人（2021）指出，按方差递增对变量进行排序通常能得到接近因果顺序的排序，并引入变量排序能力（var-sortability）来量化这种对齐程度。由于方差递增可能不切实际且具有尺度依赖性，ANM数据在基准测试中常被标准化。我们证明，合成ANM数据具有另一种尺度不变的模式：变量方差中可解释的部分（由决定系数$R^2$表征）倾向于沿因果顺序递增。这一结果导致高$R^2$排序能力（$R^2$-sortability），即按$R^2$递增对变量排序能得到接近因果顺序的排序。我们提出一种名为$R^2$-SortnRegress的高效基线算法，该算法利用高$R^2$排序能力，能够匹配甚至超越现有因果发现算法的性能。我们从理论上证明，足够大的边权重会导致噪声贡献沿因果链相对减小，从而产生越来越确定性的关系和高$R^2$。我们针对不同模拟参数刻画了$R^2$排序能力，发现在常见设定中其值较高。我们的发现揭示出高$R^2$排序能力是数据生成过程中的一种假设，与因果发现相关且隐含在许多ANM采样方案中。由于其在现实数据中的普遍性尚不可知，这种假设应当被明确说明。为助力因果发现基准测试，我们在CausalDisco库（https://causaldisco.github.io/CausalDisco/）中实现了$R^2$排序能力、$R^2$-SortnRegress算法以及ANM模拟程序。