Foundation Models for Partial Causal Identification

Alexis Bellot; Anish Dhir

cs.AIAug 21, 2026

Foundation Models for Partial Causal Identification

Authors: Alexis Bellot, Anish Dhir

Organizations: Independent · University College London

Abstract

This paper investigates the development of causal foundation models for bounding the effect of interventions and counterfactuals from observational data. We show that a canonical prior can be defined with full support over the space of structural causal models with discrete observables. With this canonical prior, we translate the problem of bounding counterfactuals into that of learning distributions over functions that map data (and possibly structural assumptions) to a causal query of interest. This extends the promising causal foundational modelling paradigm to the estimation of partially-identifiable causal effects, i.e., under unobserved confounding, where multiple values are equally compatible with the observed data and prior structural assumptions.

Figures & tables

Figure 1: Predicted posterior distributions for interventional and counterfactual probabilities in a two-variable binary SCM experiment with 100 data samples. Dashed vertical lines indicate the corresponding analytical lower and upper bounds (given by $P(x,y)\leq P(y_{x})\leq P(x,y)+P(x^{\prime})$ and $0\leq P(y_{x},\ y^{\prime}_{x^{\prime}})\leq P(x,y)+P(x^{\prime},y^{\prime})$ ).

Model	$n$	Cov.	Width	Inf. time (ms)
Gibbs	10	1.00	0.92	1053.2
	50	1.00	0.77	1095.2
	100	1.00	0.71	1146.5
	200	1.00	0.67	1322.2
	400	1.00	0.63	1521.1
CFM	10	0.99	0.70	4.2

Table 1: Coverage, mean interval width, and end-to-end inference time across context sizes for all evaluated models, averaged over 100 runs.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 2: Predicted posterior for the interventional query $P(Y_{x=1}=1)$ and the counterfactual query $P(Y_{x=1}=0,Y_{x=0}=0)$ as a function of the number of context samples $n$ for a randomly drawn SCM.

Model	$n$	Cov.	Width	Inf. time (ms)
Gibbs	10	1.000	0.854757	822.981473
	20	1.000	0.790407	839.420002
	50	1.000	0.706258	842.181667
	100	1.000	0.660208	822.117917
	200	1.000	0.623506	951.234029
	400	1.000	0.601055	1062.213033

Appendix

Table 2: Coverage, mean interval width, and end-to-end inference time across context sizes for all evaluated models, averaged over 100 runs.

Figure 3: Predicted posterior distributions for $\Psi:=P(Y_{x}=1)$ at $x=0$ (left) and $x=1$ (right) in the three-variable binary SCM experiment. Dashed vertical lines indicate the corresponding analytical lower and upper bounds ( Balke and Pearl, 1994 ) .

Parameter	Value	Description
$d$	$64$	Model (embedding) dimension
$L$	$3$	Number of alternating attention blocks
$H$	$4$	Number of attention heads
$B$	$50$	Number of histogram bins
MLP depth	$2$	Depth of predictor MLP
FF dim	$4d=256$	Feed-forward hidden dimension

Appendix

Table 3: Model and training hyperparameters.

Causal Foundation Models

Causal foundation models are pretrained models that discover causal graphs or estimate effects on new datasets using data and causal assumptions.

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Partial Identification under Causal Orders by Linear Programming

Eric Rossetto, Alessandro Antonucci

Istituto Dalle Molle di Studi sull’Intelligenza Artificiale (IDSIA) Scuola Universitaria Professionale della Svizzera Italiana (SUPSI) Lugano, Switzerland

Partial Identification Causal Inference Causal Identifiability Structural Causal Models

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Non-parametric (partial) identification of counterfactual queries typically relies on a fully specified causal graph. Motivated by settings with incomplete domain knowledge, we challenge this requirement by leveraging structural assumptions that are inherently implied by the query itself. We show that any counterfactual inquiry induces a, mostly partial, topological ordering over relevant variables, which, in turn, enables an explicit query parametrisation reducing the identification task to a linear program. This allows bounding arbitrary counterfactual and nested counterfactual queries. Our work can be viewed as a generalisation of the classical bounding framework of Tian and Pearl (2000), originally developed for probabilities of causation. We also prove the tightness of our bounds by constructing structural causal models that attain the bounds whilst being compatible with both the observed data and the query-implied order. To assess both the generality and practical utility of the proposed bounding procedure, we revisit several case studies from the literature, demonstrating how the derived bounds can be used to yield informative insights even in the absence of an input causal graph.

Abstract

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