in additionwhile computing bayesian credible interval poses no conceptual difficultiesit is not simple to derive confidence interval for the maximum likelihood estimate of thbecause the likelihood as function of th is not differentiable at th as result of this lack of smoothnessthe usual confidence intervals based on the normal approximation cannot be used we now return to the unsupervised learning setting of section but consider this from bayesian perspective recall from ( that the kullback-leibler risk for an approximating function is `(gf ( )[ln ( ln ( )dt where denotes the test data since ( ln ( dt plays no role in minimizing the riskwe consider instead the cross-entropy riskdefined as `(gf ( ln ( dt note that the smallest possible cross-entropy risk is `nf ( ln ( dt the expected generalization risk of the bayesian learner can then be decomposed as ( tn ( dt dtn ( ln `(gtn ` (tn ln ( tn ( tn { { "biascomponent "variancecomponent where gtn ( ( tn ( thg(th tn dth is the posterior predictive density after observing tn assuming that the sets tn and tn are comprised of iid random variables with density we can show (exercise that the expected generalization risk simplifies to `(gtn ln (tn ln ( )( where (tn and ( are the prior predictive densities of tn and respectively let thn argmaxth (th tn be the map estimator of th:argmaxth ln ( thassuming that thn converges to th(with probability oneand ln (tn thn ln ( tho( / )we can use the following large-sample approximation of the expected generalization risk theorem approximating the bayesian cross-entropy risk for the expected cross-entropy generalization risk satisfiese`(gtn - ln (tn ln ( where (with the dimension of the parameter vector th and thn the map estimator) ln (tn ln (tn thn ln (