# Generalised Reichenbachian Common Cause Systems

## TL;DR

This paper develops a probabilistic model for systems of multiple common causes to explain deviations in correlations, extending Reichenbach's principle of the common cause.

## Contribution

It introduces a general model for multiple common causes and establishes conditions for their existence, extending the classical principle.

## Key findings

- Two models of common cause systems are elaborated.
- Necessary and sufficient conditions for model existence are derived.
- The extended principle explains positive correlation deviations.

## Abstract

The principle of the common cause claims that if an improbable coincidence has occurred, there must exist a common cause. This is generally taken to mean that positive correlations between non-causally related events should disappear when conditioning on the action of some underlying common cause. The extended interpretation of the principle, by contrast, urges that common causes should be called for in order to explain positive deviations between the estimated correlation of two events and the expected value of their correlation. The aim of this paper is to provide the extended reading of the principle with a general probabilistic model, capturing the simultaneous action of a system of multiple common causes. To this end, two distinct models are elaborated, and the necessary and sufficient conditions for their existence are determined.