The resource theory of causal influence and knowledge of causal influence

TL;DR

This paper develops a resource-theoretic framework for quantifying and comparing causal influence between two variables, addressing both known and uncertain functional dependencies, with complete characterizations for binary variables.

Contribution

It introduces a novel resource theory for causal influence, providing methods to determine resource convertibility and a complete set of monotones, especially for binary variables.

Findings

Decidable convertibility of causal resources using monotones

Complete set of monotones for binary variable case

Linear programming approach for uncertain functional dependence

Abstract

Understanding and quantifying causal relationships between variables is essential for reasoning about the physical world. In this work, we develop a resource-theoretic framework to do so. Here, we focus on the simplest nontrivial setting -- two variables that are causally ordered, meaning that the first has the potential to influence the second, without hidden confounding. First, we introduce the resource theory that directly quantifies causal influence of a functional dependence in this setting and show that the problem of deciding convertibility of resources and identifying a complete set of monotones has a relatively straightforward solution. Following this, we introduce the resource theory that arises naturally when one has uncertainty about the functional dependence. We describe a linear program for deciding the question of whether one resource (i.e., state of knowledge about the functional dependence) can be converted to another. Then, we focus on the case where the variables are binary. In this case, we identify a triple of monotones that are complete in the sense that they capture the partial order over the set of all resources, and we provide an interpretation of each.