Causal Abstraction with Soft Interventions

TL;DR

This paper extends the theory of causal abstraction to include soft interventions, allowing for more flexible modeling of causal systems at different levels of detail.

Contribution

It generalizes $ au$-abstraction to soft interventions, defines a unique intervention mapping, and proves the explicit form of this map.

Findings

Generalization of $ au$-abstraction to soft interventions

Definition of a unique soft abstraction map

Proof of the explicit form of the intervention map

Abstract

Causal abstraction provides a theory describing how several causal models can represent the same system at different levels of detail. Existing theoretical proposals limit the analysis of abstract models to "hard" interventions fixing causal variables to be constant values. In this work, we extend causal abstraction to "soft" interventions, which assign possibly non-constant functions to variables without adding new causal connections. Specifically, (i) we generalize $\tau$-abstraction from Beckers and Halpern (2019) to soft interventions, (ii) we propose a further definition of soft abstraction to ensure a unique map $\omega$ between soft interventions, and (iii) we prove that our constructive definition of soft abstraction guarantees the intervention map $\omega$ has a specific and necessary explicit form.