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Causal Abstraction Inference under Lossy Representations
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Introduces projected abstractions that extend causal abstraction to lossy representations violating the abstract invariance condition, with graphical criteria (projected C-DAG) and identification via high-level ID, validated with GAN-NCM experiments on colored MNIST.
Problem
Existing causal abstraction definitions are ill-defined when lossy abstraction functions violate the abstract invariance condition (AIC), and cross-layer causal inference over such abstractions from limited low-level data lacks graphical criteria.
Approach
Define projected abstractions as partial SCM projections of a tau-abstraction: lost low-level information is pushed into new exogenous variables, soft interventions map high-level interventions to low-level distributions, and a partially projected C-DAG gives graphical constraints so that tau-identification reduces to classical identification on the high-level model.
Key results
- On colored MNIST estimation of P(y_x | z), the projected C-DAG GAN-NCM achieves decreasing error with more samples while the regular C-DAG model fails to learn the correct query.
- Projected sampling reproduces L1, L2, and L3 image queries even with a binary (1-bit) representation, whereas the RNCM with a binary representation lacks capacity to generate images.
- Worked insurance example gives identified effect P(Y_{X_H=x_C}=1)=0.596 via backdoor adjustment on the high-level model.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use a projected C-DAG instead of a regular C-DAG | GAN-NCM for Colored MNIST · Colored MNIST query estimation | query estimation error | Helped |
| use a projected C-DAG instead of modeling the low-level data without abstractions | GAN-NCM for Colored MNIST · Colored MNIST query estimation | query estimation error | Helped |
| use projected sampling with a binary representation instead of RNCM with a binary representation | causal colored MNIST image reconstruction · Colored MNIST causal queries | image query reproduction | Helped |
| use projected sampling instead of the non-causal conditional model | causal colored MNIST image reconstruction · Colored MNIST interventional and counterfactual queries | image query reproduction | Helped |
| use projected sampling instead of RNCM with a 16-dimensional representation | causal colored MNIST image reconstruction · Colored MNIST causal queries | image query reproduction | Mixed |
Surprising findings
- The standard C-DAG constraints are insufficient under AIC violations and can imply unsound independences, e.g., implying P(Y_x|z)=P(Y_x).
Limitations
- Partial SCM projections are not unique in induced PCH distributions, so the choice of projection/soft intervention must be specified.
- Experiments are limited to small synthetic colored MNIST settings.
Experimental setup
Datasets: Colored MNIST
Benchmarks: Colored MNIST causal queries
Metrics: mean absolute error 95% confidence intervals
Baselines: abstractionless GAN-NCM C-DAG GAN-NCM RNCM non-causal conditional model
Builds on: constructive abstraction functions C-DAG neural causal models representational NCM Pearl causal hierarchy
Closest papers by meaning
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- The Abstraction Gap in Vision-Language Causal Reasoning (ICML 2026 · similarity 0.88)
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