Abstract

In recent years part of the literature on probabilistic causality concerned notions stemming from Reichenbach’s idea of explaining correlations between not directly causally related events by referring to their common causes. A few related notions have been introduced, e.g. that of a “common cause system” (Hofer-Szabó and Rédei in Int J Theor Phys 43(7/8):1819–1826, 2004) and “causal (N-)closedness” of probability spaces (Gyenis and Rédei in Found Phys 34(9):1284–1303, 2004; Hofer-Szabó and Rédei in Found Phys 36(5):745–756, 2006). In this paper we introduce a new and natural notion similar to causal closedness and prove a number of theorems which can be seen as extensions of earlier results from the literature. Most notably we prove that a finite probability space is causally closed in our sense iff its measure is uniform. We also present a generalisation of this result to a class of non-classical probability spaces.

Highlights

  • The so-called Principle of the Common Cause is usually taken to say that any surprising correlation between two factors which are believed not to directly influence one another is due to their common cause

  • Definition 8 (Causalclosedness) We say that a classical probability space is causally up-to-n-closed w.r.t. to a relation of independence Rind if all pairs of correlated events independent in the sense of Rind possess a proper statistical common cause or a proper statistical common cause system of size at most n

  • The main result of this paper is that in finite classical probability spaces with the uniform probability measure all correlations between logically independent events have an explanation by means of a common cause or a common cause system of size 3

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Summary

Introduction

The so-called Principle of the Common Cause is usually taken to say that any surprising correlation between two factors which are believed not to directly influence one another is due to their (possibly hidden) common cause. It was natural for corresponding notions of causal closedness to be introduced; a probability space is said to be causally n-closed w.r.t. a relation of independence. Rind iff it contains an SCCS of size n for any correlation between A, B such that. We require (following Gyenis and Redei) of a causally closed probability space that all correlations be explained by means of proper—that is, differing from both correlated events by a non-zero measure event—statistical common causes. In this paper we briefly consider other independence relations, and a generalisation of our results to finite non-classical probability spaces

Preliminary Definitions
Summary of Results
Some Useful Parameters
Proof of Theorem 1
Examples
Proofs of Lemmas 1–3
Other Independence Relations
A Slight Generalisation
Findings
Conclusions and Problems
Full Text
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