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Deep Structural Causal Models for Tractable Counterfactual Inference

Nick Pawlowski, Daniel C. Castro, Ben Glocker · Imperial College London · 2020-06 · arXiv:2006.06485

Why it matters

Builds counterfactual image generation on normalizing flows and variational inference rather than diffusion, because the abduction-action-prediction procedure counterfactuals require needs a tractable invertible model. A different lineage from the rest of this section, chosen for causal correctness rather than sample quality.

What this paper does

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Before this

Structural causal models existed in econometrics and epidemiology, but typically with simple linear mechanisms. Deep generative models such as VAEs, GANs, and normalizing flows captured associations well, and some prior work reached the intervention level of Pearl's hierarchy, but the paper states that existing deep causal methods "lack tractable abduction capabilities and therefore cannot generate counterfactuals."

The problem

Build a structural causal model with deep learning components that can answer all three levels of Pearl's ladder of causation, association, intervention, and counterfactual, on high-dimensional data such as images, which requires tractably inferring the exogenous noise that abduction depends on.

The idea

Give each causal mechanism an invertible core, either a full normalizing flow or a lower-dimensional invertible transformation wrapped around a flexible non-invertible network trained with variational inference, so that abduction, inferring the noise behind an observation, becomes exact or tractably approximate rather than intractable.

How it works

This is not a diffusion model. It is a normalizing-flow and variational-inference based causal model, and its mechanism should be described on those terms. Three mechanism types are proposed. The fully invertible normalizing-flow version operates directly in data space and is exact but expensive for high-dimensional images. The amortized version the paper favors splits each mechanism into an invertible low-level component and a non-invertible high-level component, trained with an ELBO through a VAE-style encoder; the paper's stated reason is architectural flexibility, that the "invertible transformation can be made shallower, while the upstream non-invertible component... is expected to capture more of the high-level structure of the data." The choice of an invertible mechanism at all, rather than a purely generative model like a plain VAE or GAN, is explicitly because counterfactual abduction-action-prediction needs to invert the generative process to recover the exogenous noise; the paper is explicit that this tractable inversion is "missing from existing deep causal learning methods," which is its own stated reason for the whole architectural family, not merely for a preferred variant. Counterfactual inference itself follows three steps, abduction recovers the noise posterior from an observation, action edits the structural assignment for the intervened variable, and prediction resamples through the modified graph using the recovered noise.

Evidence

The paper reports no downstream clinical task and no reader study; evidence is log-likelihood, density matching against known ground truth, and qualitative visual inspection of counterfactuals. On synthetic Morpho-MNIST with a known causal graph, the causally structured model raises the causal conditional log p(intensity given thickness) to -4.30 against -5.19 for both an independent model and a plain conditional decoder, evidence the causal structure is being used rather than just correlated features. On real brain MRI from the UK Biobank, 13,750 2D slices at 64x64, the paper reports that learned joint densities of age, brain volume, and ventricle volume given sex closely match kernel density estimates of the real data, and that counterfactual interventions on age, brain volume, ventricle volume, and sex produce the expected direction of anatomical change, larger ventricles and reduced brain volume with older intervened age, for instance, while preserving cortical folding identity in generated images. This is distributional and qualitative evidence, not a clinical judgment of realism.

Limitations

The paper concedes the framework requires all variables to be observed to compute a counterfactual, limiting applicability when some causes are unmeasured. It concedes the causal graph itself is assumed rather than learned or discovered from data. It concedes that flexible neural mechanisms may not cleanly separate the causal roles of their inputs as intended, and that counterfactuals under such flexible models may not be uniquely defined. It explicitly cautions against using the framework for consequential decisions such as healthcare or policy without validating its causal assumptions.

Why it matters

It establishes that answering a genuinely causal "what if" question about an image requires invertible machinery for abduction, which conditional generation, however good its sample quality, structurally cannot provide.

Abstract, in the authors' own words

We formulate a general framework for building structural causal models (SCMs) with deep learning components. The proposed approach employs normalising flows and variational inference to enable tractable inference of exogenous noise variables - a crucial step for counterfactual inference that is missing from existing deep causal learning methods. Our framework is validated on a synthetic dataset built on MNIST as well as on a real-world medical dataset of brain MRI scans. Our experimental results indicate that we can successfully train deep SCMs that are capable of all three levels of Pearl's ladder of causation: association, intervention, and counterfactuals, giving rise to a powerful new approach for answering causal questions in imaging applications and beyond. The code for all our experiments is available at https://github.com/biomedia-mira/deepscm.

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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