proofto show ( )we apply theorem to ln -nrn (thg(thdthwhere ln (xi th- ln ( th= (thrn (th:ln (tn thn = this gives (with probability onez ln (tn thg(thdth -nr(thln( taking expectations on both sides and using nr(thne[rn (thn ) ( )we deduce ( to demonstrate ( )we derive the asymptotic approximation of ln ( by repeating the argument for ( )but replacing with nwhere necessary thuswe obtaine ln ( - nr(thp ln( then( follows from the identity ( model evidence the results of theorem have two major implications for model selection and assessment first( suggests that ln (tn can be used as crude (leading-orderasymptotic approximation to the expected generalization risk for large and fixed in this contextthe prior predictive density (tn is usually called the model evidence or marginal likelihood for the class since the integral (tn thg(thdth is rarely available in closed formthe exact computation of the model evidence is typically not feasible and may require monte carlo estimation methods secondwhen the model evidence is difficult to compute via monte carlo methods or otherwise( suggests that we can use the following large-sample approximation- ln (tn - ln (tn thn ln(nbayesian information criterion ( the asymptotic approximation on the right-hand side of ( is called the bayesian information criterion (bicwe prefer the class with the smallest bic the bic is typically used when the model evidence is difficult to compute and is sufficiently larger than for fixed pand as becomes larger and largerthe bic becomes more and more accurate estimator of - ln (tn note that the bic approximation is valid even when the true density the bic provides an alternative to the akaike information criterion (aicfor model selection howeverwhile the bic approximation does not assume that the true model belongs to the parametric class under considerationthe aic assumes that thusthe aic is merely heuristic approximation based on the asymptotic approximations in theorem although the above bayesian theory has been presented in an unsupervised learning settingit can be readily extended to the supervised case we only need to relabel the training set tn in particularwhen (as is typical for regression modelsthe training responses yn are considered as random variables but the corresponding feature vectors xn are viewed as being fixedthen tn is the collection of random responses { yn alternativelywe can simply identify tn with the response vector [ yn ]we will adopt this notation in the next example