Internally-standardized SCMs (iSCMs)

• Internally-standardized SCMs are causal models that recursively standardize each variable to have zero mean and unit variance immediately after its structural assignment.
• They effectively eliminate varsortability and R²-sortability artifacts, ensuring that synthetic datasets better mimic real-world data without artificial scale biases.
• This recursive internal standardization supports robust benchmarking of causal discovery algorithms by decoupling empirical variable scale from topological ordering.

Internally-standardized @@@@1@@@@ (iSCMs) constitute a class of SCMs in which each variable is standardized to have zero mean and unit variance during the generative process, immediately after its structural assignment and prior to being used as input to downstream mechanisms. This process decouples empirical variable scale and position in the topological order, producing synthetic datasets that avoid the artifacts of varsortability and R²-sortability endemic to standard SCM sampling schemes. iSCMs are used predominantly for benchmarking causal discovery algorithms under conditions that more accurately represent real-world data, where such sorting artifacts are absent or uninformative (Herman et al., 21 Mar 2025, Ormaniec et al., 2024).

1. Formal Definition and Generative Mechanism

Given a DAG $\mathcal{G} = (V, E)$ on $N$ vertices and a (possibly random) topological order $1 < 2 < \cdots < N$, a linear Gaussian iSCM is specified by the recursion: $$X_i := \sum_{j<i} a_{ji} X_j + U_i, \qquad U_i \sim \mathcal{N}(0, s_i^2)$$ where the raw coefficients $a_{ji}$ (for each parent $j<i$ of $i$) are drawn, typically, from a symmetric interval excluding zero (e.g., $\mathcal{U}([-2,-0.5]\cup[0.5,2])$), and $s_i = 1$.

For iSCMs, internal standardization proceeds as follows:

1. Simulate a large Monte Carlo sample ($P \gg N$) of current variables up to node $i$.
2. For each simulated draw, compute the unstandardized assignment for $X_i$.
3. Estimate the empirical standard deviation $\sigma_i$ of these draws.
4. Normalize: update $a_{ji}$ and $s_i$ to $\hat{a}_{ji}=a_{ji}/\sigma_i$ and $\hat{s}_i = 1/\sigma_i$, yielding standardized outputs $\hat{X}_i = (X_i - \overline{X}_i)/\sigma_i$ (Herman et al., 21 Mar 2025).

The result is a recursively defined process where every observed variable has unit variance, and, critically, each parent presented to the next mechanism also has unit variance at the time of input (Ormaniec et al., 2024).

For a generalized model on $d$ variables, using possibly nonlinear functions $f_i$ and arbitrary noise distributions, the iSCM recursion at node $v_i$ is: $$x_i = f_i(\{\tilde{x}_j : v_j \to v_i\}, \varepsilon_i)$$

$$\tilde{x}_i = \frac{x_i - \mathbb{E}[x_i]}{\sqrt{\operatorname{Var}[x_i]}}$$

with observed variables $\tilde{x}_i$ always having zero mean and unit variance.

2. Sortability Artifact Suppression

Varsortability

Varsortability is measured as the fraction of edges $(j \to i) \in E$ where $\operatorname{Var}(X_j) < \operatorname{Var}(X_i)$. Standard SCM sampling exhibits strong positive varsortability due to variance accumulation along the DAG; standard post-hoc normalization cannot remove this structural artifact (Herman et al., 21 Mar 2025, Ormaniec et al., 2024). iSCMs enforce $\operatorname{Var}(\tilde{X}_i) = 1$ for every $i$, eliminating systematic variance drift: varsortability is precisely $0.5$ (random) in expectation.

R²-sortability

R²-sortability is the tendency of the $R^2$—the fraction of variance explained when regressing $X_i$ on all other variables—to systematically rise with topological index. In standard linear-Gaussian SCMs, this artifact is strong when weight magnitudes exceed $1$ due to accumulated explained variance downstream. iSCMs, by standardizing parent inputs at each step and drawing raw weights i.i.d., disrupt this trend: empirical R²-sortability is approximately $0.5$ (no order), with only mild reverse R²-sortability appearing for dense graphs (Herman et al., 21 Mar 2025, Ormaniec et al., 2024).

3. Consequences for Identifiability and Causal Discovery

Linear iSCMs diverge sharply from post-hoc standardized SCMs in terms of distributional identifiability and the scaling properties of cause-explained variance.

• Bounded explained variance: For any indegree $m$ and maximum magnitude $W_{\max}$, the explained fraction of variance is bounded strictly below $1$, i.e.,

$$R^2_i \leq 1 - \frac{\sigma^2}{m^2 W_{\max}^2 + \sigma^2}$$

independent of graph depth. This ensures that even in large, deep DAGs, variables remain non-deterministic (Ormaniec et al., 2024).

• Non-identifiability beyond the Markov equivalence class: On forest-structured DAGs with Gaussian noise and known support/ordering of weights, the observational distribution generated by a linear iSCM cannot distinguish between DAGs in the same Markov equivalence class (MEC). The covariance structure is invariant under orientation flips along undirected edges, provided weights and input standardizations are maintained (Ormaniec et al., 2024). By contrast, certain post-hoc standardized SCMs may become partially identifiable with appropriate weight priors.

A plausible implication is that iSCMs yield benchmarks in which the difficulty of edge orientation better reflects the inherent Markov equivalence class ambiguity rather than secondary scale artifacts.

4. Comparison to Other SCM Benchmark Generators

A range of SCM generation methodologies are discussed in the literature, each with distinct impacts on artifact formation:

Method	Standardization	Varsortability	R²-sortability
UVN (Unit-Variance Noise, e.g., NOTEARS)	None	Strongly positive (upward)	Strongly positive (upward)
IPA (Independent Parents Assumption)	Rescales by norm	Weak, but reversed	Strongly reverse
50-50 (Squires et al. 2022)	Rescales noise/signal	None (flat)	Very strong reverse
DaO (DAG-Onion)	Distributional	Strongly reverse	R²-pattern depends on density
iSCM (internally-standardized)	Recursive internal	None (random, 0.5)	Mostly flat, mild reverse if dense

iSCM notably reduces the var- and R²-sortability artifacts most exploited by algorithms such as NOTEARS. Benchmark experiments show that using iSCMs alters the apparent ranking and behavior of causal discovery methods, sometimes substantially (Herman et al., 21 Mar 2025).

5. Implementation Details and Limitations

The practical realization of iSCMs relies on sample-based standardization:

• Coefficient draw: The raw edge weights are sampled as for UVN (e.g., uniform on $[-2,-0.5]\cup[0.5,2]$), noise variances initialized at $1$.
• Standardization sample size: Monte Carlo sample size $P$ must be large relative to $N$ (e.g., $P \in [500,2000]$ for $N \leq 20$) to stabilize estimates of $\sigma_i$, as finite sample noise induces minor residual sortability effects (Herman et al., 21 Mar 2025).
• Computational complexity scales as $O(NP)$; the technique is generally tractable for $N \leq 20$.
• Linear Gaussian assumption: Current iSCMs are designed for linear, additive, Gaussian mechanisms; nonlinear or non-Gaussian iSCMs would require revised standardization.
• Loss of real-world scale differences: Since all marginal variances are constrained to $1$, iSCMs eliminate both artificial and genuine variance heterogeneity.

Post-hoc standardization of SCM outputs does not replicate the scale invariance of internal standardization; iSCMs guarantee that each causal mechanism operates solely on unit variance inputs, which is not true post-hoc (Ormaniec et al., 2024).

6. Applications and Broader Impact

Beyond benchmarking, iSCMs have potential applications in other domains:

• Physical and engineering modeling: By ensuring unit covariance and scale-free input distributions at each mechanism, iSCMs mirror invariances present in physical systems where functional relations should not depend on measurement scale.
• Time series and large networks: Standard SCMs with large depth become nearly deterministic in downstream nodes; iSCMs maintain stable noise at every level (Ormaniec et al., 2024).
• Bayesian causal modeling: Imposing priors or regularizations on functional mechanisms is more principled when all inputs share a standard scale.
• Causal inference robustness: Internal standardization removes spurious correlations between topological order and variance/noise, possibly yielding causal effect estimates less tied to specific graph positions.

A plausible implication is that iSCMs, by decoupling statistical dependencies from artifact-inducing scale effects, provide both benchmarking and modeling regimes that more accurately reflect the challenges of causal inference in genuine observational data (Herman et al., 21 Mar 2025, Ormaniec et al., 2024).