Stochastic dominance under Bayesian learning

The concept of first-order stochastic dominance defined on distributions is inadequate in models with learning. We extend this concept to the space of distributions on distributions. We discuss conditions under which for all common observations one person's beliefs (over a set of probability distributions) dominate another person's beliefs by first-order stochastic dominance. We obtain sufficient conditions for this partial order and show that the sufficient conditions are necessary, provided that the underlying distributions satisfy an additional assumption. These conditions can be verified without taking any observations. Applications are discussed.