---
title: Internally-standardized SCMs (iSCMs)
url: https://www.emergentmind.com/topics/internally-standardized-scms-iscms
type: topic
---

# Internally-standardized SCMs (iSCMs)

Internally-standardized structural causal models (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 [2503.17037, 2406.11601].

## 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 $\hat{X}_i = (X_i - \overline{X}_i)/\sigma_i$ [2503.17037].

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 [2406.11601].

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 [2503.17037, 2406.11601]. 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 [2503.17037, 2406.11601].

## 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 [2406.11601].

- **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 [2406.11601]. 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 [2503.17037].

## 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 [2503.17037].
- **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 [2406.11601].

## 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 [2406.11601].
- **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 [2503.17037, 2406.11601].

Source: https://www.emergentmind.com/topics/internally-standardized-scms-iscms