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Multi-Level Causal Embeddings

Willem Schooltink, Fabio Massimo Zennaro

TL;DR

By defining a multi-resolution marginal problem, this paper showcases the relevance of causal embeddings for both the statistical marginal problem and the causal marginal problem; furthermore, its practical use in merging datasets coming from models with different representations is illustrated.

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.

Multi-Level Causal Embeddings

TL;DR

By defining a multi-resolution marginal problem, this paper showcases the relevance of causal embeddings for both the statistical marginal problem and the causal marginal problem; furthermore, its practical use in merging datasets coming from models with different representations is illustrated.

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.
Paper Structure (42 sections, 7 theorems, 19 equations, 4 figures, 1 algorithm)

This paper contains 42 sections, 7 theorems, 19 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contribution.
  3. Related works
  4. Preliminaries
  5. Notation.
  6. Causality
  7. Structural Causal Models.
  8. Pearl's Causal Hierarchy.
  9. Notation.
  10. Causal Abstractions
  11. Functional.
  12. Graphical.
  13. (Causal) Marginal Problem
  14. SCM Projections
  15. Embeddings
  16. ...and 27 more sections

Key Result

Lemma 1

Def.def:alpha-emb in terms of projections is equivalent to Def.def:alpha-emb-alt in terms of explicit graphical constraints.

Figures (4)

  • 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 $X\rightarrow Y$.
  • Figure 2: A high-level causal model of a simplified ecosystem.
  • Figure 3: Two low-level causal models, each modeling a sub-system of a simplified ecosystem.
  • 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.

Theorems & Definitions (39)

  • Definition 1: Structural Causal Model
  • Definition 2: Intervention
  • Definition 3: $\alpha$-abstraction
  • Definition 4: $\mathcal{L}_i$-Abstraction error
  • Definition 5: Cluster DAG
  • Definition 6: Graphical $\mathcal{L}_i$-Consistency
  • Definition 7: Marginal Problem
  • Definition 8: Causal Marginal Problem
  • Definition 9: SCM Projection
  • Example 1: Simplified Ecosystem Modeling
  • ...and 29 more