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Diffusion Causal Models for Counterfactual Estimation

Pedro Sanchez, Sotirios A. Tsaftaris · University of Edinburgh · 2022-02 · arXiv:2202.10166

Why it matters

The first diffusion-based deep structural causal model: forward diffusion for abduction, then anti-causal gradient guidance through the reverse process for action and prediction. Its experiments are MNIST and ImageNet only, with no medical images, so it belongs here as the methodological bridge that brought diffusion into the causal counterfactual family rather than as a clinical result.

What this paper does

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Before this

Deep structural causal models had been demonstrated on semi-synthetic, low-dimensional data, and quantifying the causal effect of interventions for high-dimensional images with neural networks remained an open problem. Separately, score-based and energy-based generative models were emerging as an alternative to GANs and VAEs for image synthesis, but had not been connected to Pearl's abduction-action-prediction framework.

The problem

How to perform counterfactual inference (abduction, action, prediction) over high-dimensional images using a single generative mechanism, given a known causal graph relating an image to a small set of label variables.

The idea

Treat the diffusion process itself as the causal mechanism: deterministic forward diffusion recovers the exogenous noise of the image (abduction), and an intervention is carried out by guiding the reverse diffusion process with the gradient of an anti-causal classifier trained on the intervened variable (action and prediction), unifying stochastic differential equation diffusion with structural causal models.

How it works

A DDIM forward pass deterministically encodes the image into a latent noise variable. Reverse diffusion is then guided by the gradient, with respect to the image, of an anti-causal predictor for the intervened parent variable, pushing the sample toward the requested intervention. The paper introduces Counterfactual Latent Divergence (CLD), a metric that scores the minimality of a counterfactual via KL divergence in the latent space of a class-conditional autoencoder, balancing proximity to the original class against distance from the target class. A scale hyperparameter controlling guidance strength is tuned by optimizing CLD (optimal value 0.7 on MNIST).

Evidence

Distribution and reconstruction metrics only, no causal axiom tests, no downstream task utility, and no reader study. Quantitative results are reported only on MNIST: IM1 0.94±0.02 versus 1.10±0.03 for a Looveren & Klaise baseline, IM2 0.04±0.00 versus 0.05±0.00, and CLD 1.08±0.03 versus 1.25±0.03. ImageNet 256x256 is used only for qualitative visual examples, with no quantitative table. No medical imaging dataset appears anywhere in the experiments, confirming the atlas's filing of this paper as a methodological precursor rather than a clinical result.

Limitations

The authors state the empirical setting is restricted to specifications for two variables (image and one label), and that extending to more complex causal graphs with multiple confounders, including handling graph mutilation, is left to future work.

Why it matters

This paper brought diffusion models into the causal-counterfactual family, replacing the GAN and VAE causal mechanisms of earlier deep structural causal model work with score-based classifier guidance, and supplies the abduction-action-prediction-via-diffusion machinery that later papers in this line (ribeiro-2023's successors, pcgm-2025, causal-adapter-2025) build on or diverge from. Because its own experiments never touch medical imaging, it functions in this atlas as a methodological bridge, not as clinical evidence, exactly as the existing filing states.

Abstract, in the authors' own words

We consider the task of counterfactual estimation from observational imaging data given a known causal structure. In particular, quantifying the causal effect of interventions for high-dimensional data with neural networks remains an open challenge. Herein we propose Diff-SCM, a deep structural causal model that builds on recent advances of generative energy-based models. In our setting, inference is performed by iteratively sampling gradients of the marginal and conditional distributions entailed by the causal model. Counterfactual estimation is achieved by firstly inferring latent variables with deterministic forward diffusion, then intervening on a reverse diffusion process using the gradients of an anti-causal predictor w.r.t the input. Furthermore, we propose a metric for evaluating the generated counterfactuals. We find that Diff-SCM produces more realistic and minimal counterfactuals than baselines on MNIST data and can also be applied to ImageNet data. Code is available https://github.com/vios-s/Diff-SCM.

Research line

Medical: causal counterfactualsascendant

Method note — the shared flow-matching interpolation

Every flow-matching / rectified-flow paper in this atlas trains toward a straight-line path between a noise sample x₀ and a data sample x₁:

x_t = (1-t)\,x_0 + t\,x_1, \qquad v_\theta(x_t, t) \approx x_1 - x_0

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