Estimating Causal Effects in Gaussian Linear SCMs with Finite Data

AI summary

Overview

Research area: Causal inference and machine learning — specifically causal effect estimation in Gaussian Linear Structural Causal Models (GL-SCMs) under latent confounding, using only finite observational data.

Technical level: Advanced. The paper assumes familiarity with structural causal models, do-calculus, identifiability, Gaussian graphical models, and Expectation-Maximization.

Scope: The paper introduces a restricted subclass of Gaussian linear SCMs (Centralized Gaussian Linear SCMs, or CGL-SCMs), proves it preserves identifiability of causal effects, and gives an EM-based algorithm for learning its parameters from finite samples.

Publication details: arXiv:2601.04673v1 [cs.LG], 08 Jan 2026, CC BY 4.0. Authors: Aurghya Maiti and Prateek Jain, Department of Computer Science, Columbia University, NY 10025 (equal contribution). Keywords listed: Machine Learning, ICML.

What This Paper Is About

Estimating causal effects from observational data is hard, especially when unobserved confounders are present. Gaussian Linear SCMs are popular because they are analytically convenient, but they are overparameterized — the number of free parameters is often too large relative to what finite data can pin down, making exact parameter estimation infeasible.

The paper's goal is to close the gap between theory (which typically assumes infinite data or partial knowledge of parameters) and practice (finite samples), by defining a simplified but equally expressive class of Gaussian linear SCMs whose parameters can actually be estimated from limited observational samples, and then using that estimated model to answer identifiable causal queries.

Key Contributions

1. Introduces Centralized Gaussian Linear SCMs (CGL-SCMs): a subclass of GL-SCMs in which exogenous confounders are drawn from a standard normal N(0, I) and exogenous non-confounding errors have zero mean, drastically reducing the number of free parameters.

2. Proves equal expressive power (Theorem 2.3): for any GL-SCM M′, there exists a CGL-SCM M with the same causal graph such that P^{M′}(X) = P^M(X). A companion result (Theorem 2.4) shows that if a query Q is identifiable in the causal graph G, then P^{M′}(Q) = P^M(Q) — so estimating Q in the CGL-SCM gives an estimate of Q in the original GL-SCM.

3. Develops a scalable EM algorithm (Algorithm 1, "CGL-Go") for estimating CGL-SCM parameters from finite data in the presence of latent variables, together with a second algorithm (Algorithm 2, "CGL-Edge") that recovers individual edge weights T from the aggregated influence matrix B.

4. Validates the approach experimentally on synthetic data generated from two canonical causal graphs — the Frontdoor graph and the Napkin graph — showing that recovered interventional distributions closely match the originals.

Main Findings

• Vectorized reformulation enables EM: The CGL-SCM is written as X = Bᵀμ + BᵀCᵀU + Bᵀε, where T holds direct edge weights, B = I + Σ_{i=1}^{d} T^i captures total influence along all directed paths (d is the longest path length), and C captures direct effects of exogenous confounders U on endogenous variables X. This form aligns the model with the classical linear-Gaussian framework needed by EM.

• Non-unique solutions are acceptable: The paper explicitly notes that parameter estimation may not have a unique solution. The stated goal is not to recover the true data-generating model but to find a CGL-SCM that induces the same L1 (observational) distribution; by Theorem 2.4, any identifiable interventional or counterfactual query then matches the original model.

• Graphical constraints require masking: Naively applying EM can produce B and C that violate the causal graph, so mask matrices B_m and C_m (derived from T_m, with entries set by whether a directed path or edge exists) are used to keep the optimization within the graph structure.

• Gradient ascent replaces closed form: No closed-form solution was found for B and C during maximization, so gradient ascent with masked gradients is used in the M-step; μ has a closed-form update (μ = (1/N) Σ_i [(Bᵀ)⁻¹Xⁱ − Cᵀμ_{Uⁱ|Xⁱ}]).

• Frontdoor graph recovery: With 10,000 samples generated from the frontdoor CGL-SCM (μ = [0.3, 0.1, 0.2]ᵀ), the estimator returned μ̂ = [0.3002, 0.0963, 0.2007], Ĉ = [−0.2149, 0, 0.2626] (true C = [−0.2, 0, 0.3]), and T̂ with off-diagonal entries 0.5012 and 0.9011 (true T had 0.5 and 0.9).

• Frontdoor causal distributions: P(X₃ | do(X₂=1)) was N(1.1, 1.09) originally versus N(1.1018, 1.069) estimated; P(X₃ | do(X₁=1)) was N(0.74, 1.9) originally versus N(0.7391, 1.881) estimated.

• Napkin graph recovery: With 10,000 samples from the napkin CGL-SCM (μ = [0.8, −0.9, 0.01, −0.5]ᵀ), the estimator returned μ̂ = [0.7994, −0.9044, 0.0113, −0.4894], Ĉ = [[−0.1271, 0, 0.1665, 0], [0.3389, 0, 0, 0.4113]] (true C = [[−0.2, 0, 0.1, 0], [0.3, 0, 0, 0.4]]), and T̂ with entries 0.1024, 0.8985, and 0.7945 (true T had 0.1, 0.9, and 0.8).

• Napkin causal distributions: P(X₄ | do(X₃=1)) was N(0.3, 1.16) originally versus N(0.3051, 1.1692) estimated; P(X₄ | do(X₁=1)) was N(−1.068, 2.3248) originally versus N(−0.9721, 2.3274) estimated.

• Overall experimental verdict: The authors state that for both examples the estimated causal distributions are very close to the original distributions.

Methodology in Plain English

The authors start from a standard Gaussian linear causal model in which every variable is a weighted sum of its observed parents, its unobserved confounding parents, a bias term, and a noise term. In the general version, both confounders and noises can have arbitrary means and variances, which balloons the parameter count.

Their fix is to standardize: force all confounders to be N(0, I) and all non-confounding errors to have zero mean. Any parameters lost by this restriction are absorbed into the remaining coefficients and bias terms, so nothing is lost in terms of the observational distribution. That means any causal query you could identify in the original model yields the same answer in the simplified one.

Because the model contains latent variables, they use Expectation-Maximization. To apply it, they rewrite the model in matrix form: a direct-edge matrix T, a total-influence matrix B built as I + T + T² + … up to the longest path length, and a matrix C mapping confounders to observed variables. The E-step computes the conditional Gaussian distribution of the latent confounders given each observed data point; the M-step maximizes the expected log-likelihood. Since no closed-form update exists for B and C, they use gradient ascent with gradients masked so that no edges appear where the causal graph has none. A final separate step converts the aggregated influence matrix B back into individual edge weights T.

They test on two well-known graph structures — Frontdoor and Napkin — generating 10,000 samples from a known CGL-SCM, running the estimator, and comparing the resulting interventional distributions against the ground truth. The assumption throughout is that the causal graph is known and only finite observational (L1) samples are available.

Why This Matters

Impact on research: The paper bridges two research traditions that have largely stayed separate — the non-parametric identification literature (Pearl's do-calculus and related algorithmic work, including L2 interventional and L3 counterfactual queries) and the parametric econometrics/statistics literature on linear systems. It also addresses a practical gap: prior finite-data causal effect estimation work has focused mainly on non-parametric models, and much of the linear-SCM identifiability work assumes infinite data, partial parameter knowledge, or the Markovian assumption, which rules out latent confounding.

Potential real-world application areas (not demonstrated in the paper, but implied by the setting):

• Epidemiology and public health, where treatment effects must be estimated from observational cohorts with unmeasured confounders.
• Economics and policy evaluation, where linear structural models with latent factors are common.
• Genomics and biology, where gene regulatory networks are modeled as linear systems with unobserved drivers.
• Industrial and business analytics, where A/B testing is impossible and only logged observational data from a known dependency graph exists.

Industry relevance: The method targets exactly the regime companies face — a known causal diagram (from domain experts or prior discovery), a limited sample, and latent confounders. Because the algorithm outputs a full SCM, once fitted it can answer many interventional and counterfactual queries without re-running the estimation.

Future Directions

1. More general distributions: The conclusion states that future work can explore distributions beyond the Gaussian setup.

2. Bounds for non-identifiable queries: The authors propose finding bounds for non-identifiable queries under finite data, rather than only handling queries that are exactly identifiable.

3. Uniqueness and identifiability of parameters: The paper notes that parameter estimation in this setting may not have a unique solution; characterizing which CGL-SCMs induce the same L1 distribution remains open.

4. Scaling the optimization: The M-step relies on gradient ascent with masks rather than a closed-form update for B and C, which raises the question of whether closed-form or faster optimization can be derived for larger graphs.

Target Audience

The paper is most useful to researchers and graduate students in causal inference, probabilistic graphical models, and machine learning theory who work on identifiability and estimation from finite data. It is also relevant to applied statisticians and econometricians who fit linear structural models with latent confounders, and to practitioners in fields such as epidemiology, economics, and genomics who have a known causal graph, limited observational samples, and a need for interventional or counterfactual estimates. A strong background in linear algebra, Gaussian distributions, and EM is needed to follow the derivations.

Authors’ abstract

Estimating causal effects from observational data remains a fundamental challenge in causal inference, especially in the presence of latent confounders. This paper focuses on estimating causal effects in Gaussian Linear Structural Causal Models (GL-SCMs), which are widely used due to their analytical tractability. However, parameter estimation in GL-SCMs is often infeasible with finite data, primarily due to overparameterization. To address this, we introduce the class of Centralized Gaussian Linear SCMs (CGL-SCMs), a simplified yet expressive subclass where exogenous variables follow standardized distributions. We show that CGL-SCMs are equally expressive in terms of causal effect identifiability from observational distributions and present a novel EM-based estimation algorithm that can learn CGL-SCM parameters and estimate identifiable causal effects from finite observational samples. Our theoretical analysis is validated through experiments on synthetic data and benchmark causal graphs, demonstrating that the learned models accurately recover causal distributions.