What Bayesians Expect of Each Other
M. J. Bayarri, Morris H. DeGroot · Journal of the American Statistical Association · 1991
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 θ, and study Bayesian A's expectation of Bayesian B's posterior distribution for θ given some data Y. We show that when θ 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 θ. For a vector parameter θ 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.