# A Combinatorial Solution to Causal Compatibility

## TL;DR

This paper introduces the possible worlds framework, a novel combinatorial approach for determining causal compatibility in complex systems with hidden variables, providing complete solutions and practical tests.

## Contribution

It presents the possible worlds diagram as a new graphical tool for certifying causal compatibility, especially in challenging partial information scenarios, and develops a hierarchy of tests that converges to sufficiency.

## Key findings

- Proves causal incompatibility in cases where previous techniques failed
- Provides a complete solution to the possibilistic causal compatibility problem
- Develops a hierarchy of tests that converges to sufficiency

## Abstract

Within the field of causal inference, it is desirable to learn the structure of causal relationships holding between a system of variables from the correlations that these variables exhibit; a sub-problem of which is to certify whether or not a given causal hypothesis is compatible with the observed correlations. A particularly challenging setting for assessing causal compatibility is in the presence of partial information; i.e. when some of the variables are hidden/latent. This paper introduces the possible worlds framework as a method for deciding causal compatibility in this difficult setting. We define a graphical object called a possible worlds diagram, which compactly depicts the set of all possible observations. From this construction, we demonstrate explicitly, using several examples, how to prove causal incompatibility. In fact, we use these constructions to prove causal incompatibility where no other techniques have been able to. Moreover, we prove that the possible worlds framework can be adapted to provide a complete solution to the possibilistic causal compatibility problem. Even more, we also discuss how to exploit graphical symmetries and cross-world consistency constraints in order to implement a hierarchy of necessary compatibility tests that we prove converges to sufficiency.