  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.
