8. Structure of Bayes solutions. (i) Let be an unobservable random quantity with probability density p(0), and let the probability density of X be p(x) when = 0. Then 8 is a Bayes solution of a given decision problem if for each x the decision 8(x) is chosen so as to minimize (L(0, 8(x))(0|x) do, where (0|x) = p(0) Po(x)/p(0) Pe (x) de' is the conditional (a posteriori) probabil- ity density of given x. (ii) Let the problem be a two-decision problem with the losses as given in Example 12. Then the Bayes solution consists in choosing decision do if aP{w|x}