Turning Probabilities into Expectations

Abstract

Suppose that you specify your prior probability that an unknown quantity $\theta$ lies in each member of a disjoint partition of the values of $\theta$. What does this imply about your prior mean and variance for $\theta$, and your posterior mean and variance, given sample information? We provide a partial answer by modifying a suggestion of Manski for incorporating the cost of specification of prior probabilities into the analysis of decision problems. This modification leads to a simple explicit solution in the problem of estimating the mean of a distribution, with quadratic loss, in the class of linear functions of the sample, and this solution is related to the problem of turning probabilities into expectations.