example (polynomial regression (cont )consider example once againbut now in bayesian frameworkwhere the prior knowledge on ( bis specified by ( / and ( )and is (matrixhyperparameter let :(xx - )- then the posterior can be written as (bs exp ky-xbk ( ps ) / - exp (yx ( ps / | | / ( )-( + )/ - ks- / ( ) ( exp ( )( + )/ | | / ! ( )where :sxy and : ( xsx) /( are the map estimates of and and (yis the model evidence for (yg(bs ydb ds ( + + exp | | / ( ) / | | / ( ) / + ds | | / ( / | | / ( ( ) / thereforebased on ( )we have `(gtn - ln (yn ln ( ln ( / ln |dln |son the other handthe minus of the log-likelihood of can be written as ln ( bs ky xbk ln( ps ks- / ( ) ( ln( ps thereforethe bic approximation ( is - ln ( bs ( ln(nn[ln( ( ln( ( )( where the extra ln(nterm in ( ln(nis due to the inclusion of in th ( bfigure shows the model evidence and its bic approximationwhere we used hyperparameter for the prior density of we can see that both approximations exhibit pronounced minimum at thus identifying the true polynomial regression model compare the overall qualitative shape of the cross-entropy risk estimate with the shape of the square-error risk estimate in figure