This implies that the Bayes risk is 0. The Bayes risk of δπ(X) can be calculated as repeated expectation in two ways, r(π, δπ) = EθEX|θ(θ − δπ(X)) = EXEθ|X(θ − δπ(X)). Thus, conveniently choosing either EθEX|θ or EXEθ|X and using the properties of conditional expectation we have, r(π, δπ) = EθEX|θθ2 −EθEX|θθδπ(X)−EXEθ|Xθδπ(X) + EXEθ|Xδ2 π(X) = EθEX|θθ2 −Eθθ[EX|θδπ(X)]− EXδπ(X)Eθ|Xθ + EXEθ|Xδ2 π...

Consider the problem of estimating normal means from independent observations with known variances, possibly different from each other. Suppose that a second-level normal model is specified on the unknown means, with the prior means depending on a vector of covariates and the prior variances constant. For this two-level normal model, existing empirical Bayes methods are constructed from the Bay...