Structure of Bayes solutions. (i) Let be an unobservable random quantity with probability density (), and

Structure of Bayes solutions.

(i) Let Θ be an unobservable random quantity with probability density ρ(θ), and let the probability density of X be pθ(x) when Θ = θ. Then δ is a Bayes solution of a given decision problem if for each x the decision δ(x) is chosen so as to minimize L(θ, δ(x))π(θ | x) dθ, where π(θ | x) = ρ(θ)pθ(x)/
ρ(θ

)pθ
(x) dθ
is the conditional (a posteriori) probability density of Θ given x.
(i) Let the problem be a two-decision problem with the losses as given in Example 1.5.5. Then the Bayes solution consists in choosing decision d0 if aP{Θ ∈ ω1 | x} < bP{Θ ∈ ω0 | x}
and decision d1 if the reverse inequality holds. The choice of decision is immaterial in case of equality.
(iii) In the case of point estimation of a real-valued function g(θ) with loss function L(θ, d)=(g(θ) − d)
2, the Bayes solution becomes δ(x) = E[g(Θ) | x]. When instead the loss function is L(θ,

d) = |g(θ) − d|, the Bayes estimate δ(x) is any median of the conditional distribution of g(Θ) given x.
[(i): The Bayes risk r(ρ, δ) can be written as [
L(θ, δ(x))π(θ | x) dθ] × p(x) dx, where p(x) = ρ(θ

)pθ
(x) dθ

.
(ii): The conditional expectation L(θ, d0)π(θ | x) dθ reduces to aP{Θ ∈ ω1 | x}, and similarly for d1.]