Bayesian Decision Theory and Stochastic Independence

Stochastic independence (SI) 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, hence a property that any theory on the foundations of probability should be able to account for. Bayesian decision theory, which is one such theory, appears to be wanting in this respect. In Savage's classic treatment, postulates on preferences under uncertainty are shown to entail a subjective expected utility (SEU) representation, and this permits asserting only the existence and uniqueness of a subjective probability, regardless of its properties. What is missing is a preference postulate that would specifically connect with the SI property. The paper develops a version of Bayesian decision theory that fills this gap. In a framework of multiple sources of uncertainty, we introduce preference conditions that jointly entail the SEU representation and the property that the subjective probability in this representation treats the sources of uncertainty as being stochastically independent. We give two representation theorems of graded complexity to demonstrate the power of our preference conditions. Two sections of comments follow, one connecting the theorems with earlier results in Bayesian decision theory, and the other connecting them with the foundational discussion on SI in probability theory and the philosophy of probability. Appendices offer more technical material