Bayesian Decision Theory and Stochastic Independence

TL;DR

This paper introduces preference axioms within a new framework of twofold uncertainty that explicitly entail stochastic independence, addressing a gap in Bayesian decision theory.

Contribution

It proposes a novel set of preference axioms that derive stochastic independence, expanding Bayesian decision theory beyond traditional definitional properties.

Findings

Preference axioms imply stochastic independence

Addresses a lacuna in Bayesian decision theory

Enhances understanding of twofold uncertainty

Abstract

Stochastic independence has a complex status in probability theory. It is not part of the definition of a probability measure, but it is nonetheless an essential property for the mathematical development of this theory. Bayesian decision theorists such as Savage can be criticized for being silent about stochastic independence. From their current preference axioms, they can derive no more than the definitional properties of a probability measure. In a new framework of twofold uncertainty, we introduce preference axioms that entail not only these definitional properties, but also the stochastic independence of the two sources of uncertainty. This goes some way towards filling a curious lacuna in Bayesian decision theory.