Compositional Abstraction Error and a Category of Causal Models
Interventional causal models describe 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 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 from fine-grained to coarse-grained variables. Here, we argue that compositionality is a desideratum for model transformations and the associated errors. We develop a framework for model transformations and abstractions with a notion of error that is compositional: when abstracting a reference model M modularly, first obtaining M' and then further simplifying that to obtain M'', then the composite transformation from M to M'' exists and its error can be bounded by the errors incurred by each individual transformation step. Category theory, the study of mathematical objects via the compositional transformations between them, offers a natural language for developing our framework. We introduce a category of finite interventional causal models and, leveraging theory of enriched categories, prove that our framework enjoys the desired compositionality properties.