
542 Current Trends in Bayesian Methodology with Applications
The Rao-Blackwell theorem is often use d to s how how the knowledge of
a conditional expec ted value can improve the variance of an estimate, but
this is only a special case of a more general theorem which shows that any
convex cost-function is improved using this technique. For completeness, the
Rao-Blackwell theorem is given in Theorem 26.3.1, [28, p. 51 ].
Theorem 26.3.1 (The Rao-Blackwell theorem [28]). Let x be a random ob-
servable with the distribution P
θ
∈ P = {P
θ
′
∈ Ω}, and let t be s ufficient
statistics for P
1
. Let
ˆ
θ be an estimator of an estimand g(θ), and let the loss
function L(