Exchangeability and the Law of Maturity

TL;DR

This paper investigates how finite exchangeability, a weaker assumption than infinite exchangeability, aligns with the law of maturity and related probabilistic beliefs, providing conditions and models that support this compatibility.

Contribution

It demonstrates that finite exchangeability can reconcile the law of maturity with probabilistic beliefs, unlike infinite exchangeability, and offers conditions and examples for this.

Findings

Finite exchangeability is compatible with the law of maturity.

Sufficient conditions for beliefs to hold under finite exchangeability.

Illustrations using common parametric models.

Abstract

The law of maturity is the belief that less-observed events are becoming mature and, therefore, more likely to occur in the future. Previous studies have shown that the assumption of infinite exchangeability contradicts the law of maturity. In particular, it has been shown that infinite exchangeability contradicts probabilistic descriptions of the law of maturity such as the gambler's belief and the belief in maturity. We show that the weaker assumption of finite exchangeability is compatible with both the gambler's belief and belief in maturity. We provide sufficient conditions under which these beliefs hold under finite exchangeability. These conditions are illustrated with commonly used parametric models.