4.23 ( ) www In this exercise, we derive the BIC result (4.139) starting from the Laplace approximation to the model evidence given by (4.137). Show that if the prior over parameters is Gaussian of the form p(θ) = N(θ|m,V0), the log model evidence under the Laplace approximation takes the form ln p(D)  ln p(D|θMAP) − 1 2

(θMAP −m)TV−1 0 (θMAP −m) − 1 2

ln |H| + const whereHis the matrix of second derivatives of the log likelihood ln p(D|θ) evaluated at θMAP. Now assume that the prior is broad so that V−1 0 is small and the second term on the right-hand side above can be neglected. Furthermore, consider the case of independent, identically distributed data so thatHis the sum of terms one for each data point. Show that the log model evidence can then be written approximately in the form of the BIC expression (4.139).