WHAT BAYESIANS EXPECT OF EACH OTHER

Our goal is to study general properties of one Bayesian's subjective beliefs about the behavior of another Bayesian's subjective beliefs. We consider two Bayesians, A and B, who have different subjective distributions for a parameter theta, and study Bayesian A's expectation of Bayesian B's posterior distribution for theta given some data Y. We show that when theta can take only two values, Bayesian A always expects Bayesian B's posterior distribution to lie between the prior distributions of A and B. Conditions are given under which a similar result holds for an arbitrary real-valued parameter theta. For a vector parameter theta we present useful expressions for the mean vector and covariance matrix of A's expectation of B's posterior distribution. Examples are given illustrating the relevance of the conditions under which the results are derived.