This paper explores the concept of "Causal Optimal Transport of Abstractions" (COTA), a novel approach to understanding the relationship between different causal models representing the same system at different levels of detail. COTA aims to unify various notions of interventional consistency, allowing for more flexible and robust evaluation of causal models under different assumptions. The authors propose a framework that utilizes graph-based causal models, enabling the comparison and matching of abstractions between two causal models. This approach has significant implications for the field of explainability, as it provides a means to evaluate the consistency of neural networks with respect to abstracted models. By learning an abstraction between two causal models, COTA offers a way to understand how different levels of detail can be combined and used to make more informed decisions.
Introduction
Causality is a fundamental concept in many fields, including economics, social science, and artificial intelligence. However, dealing with complex systems and multiple factors often makes it difficult to model causal relationships accurately. To address this challenge, researchers have proposed various notions of interventional consistency, which evaluate how well a causal model can explain the behavior of a system under different interventions. In this paper, we propose an extension of these ideas by introducing Causal Optimal Transport of Abstractions (COTA).
COTA: A New Framework for Interventional Consistency
The core idea behind COTA is to use optimal transport theory to compare and match abstractions between two causal models. In particular, we consider the problem of finding a map that maps the variables in one model to those in another while preserving the causal relationships between them. This allows us to define a notion of interventional consistency that is more flexible and robust than previous approaches.
Definition 1 (Interventions): Given a structural causal model M, an intervention ι = do(A = a) replaces each function fi associated with the variable Ai with the constant ai. This mutilates the graph underlying M by removing the incoming arrows in each node Ai and replacing fi with the constant ai.
Definition 2 (Abstractions): An abstraction is a map that relates two causal models representing the same system in different levels of detail. Given an SCM M, we define an abstraction PM as a probability distribution over the endogenous variables X. Whenever clear from the context, we shorthand do(A = a) to do(a).
COTA Framework
The COTA framework consists of two main components: (i) a measure of interventional consistency and (ii) a map relating two causal models representing the same system in different levels of detail. The interventional consistency measure is based on the concept of optimal transport theory, which provides a way to compare and match abstractions between two causal models.
Measure of Interventional Consistency
The interventional consistency measure is defined as the minimum distance between the probability distributions PM and PMι (X) under the transport map T. In other words, it measures how well the probability distributions in both models match given an intervention ι = do(A = a).
Map Relating Two Causal Models
The map relating two causal models representing the same system in different levels of detail is defined as follows. Given two SCMs M and Mι, we define a probability distribution PM over X and a corresponding probability distribution PMI over X, where PM(X) = PMI(X). The map T between M and Mι is then defined as:
T(a) = argmin{b | PM(X) = PMI(X)}
where b is the value of the variables in M that corresponds to a in Mι.
Results
We demonstrate the effectiveness of COTA through several examples, including one on explainability where we show how our framework can be used to evaluate the consistency of neural networks with respect to abstracted models. We also provide an analysis of the limitations of our approach and discuss future research directions.
Conclusion
In conclusion, Causal Optimal Transport of Abstractions (COTA) provides a novel approach to understanding the relationship between different causal models representing the same system at different levels of detail. By leveraging optimal transport theory, COTA offers a more flexible and robust way to evaluate interventional consistency, with significant implications for fields such as explainability. Our work provides a foundation for further research in this area, paving the way for new applications and insights into the complex relationships between causality and abstraction.
