Compositional Abstraction Error and a Category of Causal Models

TL;DR

This paper develops a formal, category-theoretic framework for understanding compositional abstraction errors in interventional causal models, enabling systematic and bounded model transformations from fine-grained to coarse-grained variables.

Contribution

It introduces a category of finite interventional causal models and proves compositionality properties of abstraction errors using enriched category theory.

Findings

Framework formalizes model transformations and error bounds.

Proves compositionality of abstraction errors.

Provides a mathematical foundation for causal model abstraction.

Abstract

Interventional causal models describe several joint distributions over some variables used to describe a system, one for each intervention setting. They provide a formal recipe for how to move between the different joint distributions and make predictions about the variables upon intervening on the system. Yet, it is difficult to formalise how we may change the underlying variables used to describe the system, say moving from fine-grained to coarse-grained variables. Here, we argue that compositionality is a desideratum for such model transformations and the associated errors: When abstracting a reference model M iteratively, first obtaining M' and then further simplifying that to obtain M'', we expect the composite transformation from M to M'' to exist and its error to be bounded by the errors incurred by each individual transformation step. Category theory, the study of mathematical objects via compositional transformations between them, offers a natural language to develop our framework for model transformations and abstractions. We introduce a category of finite interventional causal models and, leveraging theory of enriched categories, prove the desired compositionality properties for our framework.