Multi-Level Causal Embeddings

Willem Schooltink; Fabio Massimo Zennaro

cs.AIFeb 25, 2026

Multi-Level Causal Embeddings

Authors: Willem Schooltink, Fabio Massimo Zennaro

Organizations: Department of Informatics, University of Bergen

Abstract

Abstractions of causal models allow for the coarsening of models such that relations of cause and effect are preserved. Whereas abstractions focus on the relation between two models, in this paper we study a framework for causal embeddings which enable multiple detailed models to be mapped into sub-systems of a coarser causal model. We define causal embeddings as a generalization of abstraction, and present a generalized notion of consistency. By defining a multi-resolution marginal problem, we showcase the relevance of causal embeddings for both the statistical marginal problem and the causal marginal problem; furthermore, we illustrate its practical use in merging datasets coming from models with different representations.

Figures & tables

Panel of Figure 1 — Figure 1: A visual comparison between abstractions (left) and embeddings (right). Note that abstractions (blue) have mappings to all variables in the high-level model, whereas the embeddings (orange) provides a fine-grained description only of the sub-system $X\rightarrow Y$ .

Panel of Figure 2 — Figure 2: A high-level causal model of a simplified ecosystem.

Panel of Figure 3 — Figure 3: Two low-level causal models, each modeling a sub-system of a simplified ecosystem.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 4 — Figure 4: A visual comparison between the predicted estimation $\hat{P}(\text{Humans},\text{Predators})$ and the evaluation distribution $P(\text{Humans},\text{Predators})$ . Imputation allows for approximation of distributions otherwise not available.

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Causal and Compositional Abstraction

Robin Lorenz, Sean Tull

Quantinuum, London, United Kingdom

Category Theory Causal Abstraction Structural Causal Models

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Abstracting from a low level to a more explanatory high level of description, and ideally while preserving causal structure, is fundamental to scientific practice, to causal inference problems, and to robust, efficient and interpretable AI. We present a general account of abstractions between low and high level models as natural transformations, focusing on the case of causal models. This provides a new formalisation of causal abstraction, unifying several notions in the literature, including constructive causal abstraction, Q-

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consistency, abstractions based on interchange interventions, and

distributed' causal abstractions. Our approach is formalised in terms of category theory, and uses the general notion of a compositional model with a given set of queries and semantics in a monoidal, cd- or Markov category; causal models and their queries such as interventions being special cases. We identify two basic notions of abstraction: downward abstractions mapping queries from high to low level; and upward abstractions, mapping concrete queries such as Do-interventions from low to high. Although usually presented as the latter, we show how common causal abstractions may, more fundamentally, be understood in terms of the former. Our approach also leads us to consider a new stronger notion of

component-level' abstraction, applying to the individual components of a model. In particular, this yields a novel, strengthened form of constructive causal abstraction at the mechanism-level, for which we prove characterisation results. Finally, we show that abstraction can be generalised to further compositional models, including those with a quantum semantics implemented by quantum circuits, and we take first steps in exploring abstractions between quantum compositional circuit models and high-level classical causal models as a means to explainable quantum AI.

Abstract

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