laplace' approximation finallywe mention laplace' approximationwhich shows how integrals or expectations behave under the normal distribution with vanishingly small variance theorem laplace' approximation suppose that thn thwhere thlies in the interior of the open set th and that sn is covariance matrix such that sn slet th be continuous function with (th thenas - / (the (th-thn sn (th-thn dth (th| ( th proof(sketch for bounded domain th the left-hand side of ( can be written as the expectation with respect to the (thn sn /ndistributionz (th th exp (th thn ) - dth | sn (th| sn [ (xn ) {xn th}]| sn / | / th / where xn sn /nlet ( ithenth sn zn has the same distribution (thn , / as xn and thn sn zn thas by continuity of (th) {th thin the interior of thas :  / / [ (xn ) {xn th} thn snnz thn snnz th - (th) {ththsince thlies in the interior of thwe have {thth completing the proof as an application of theorem we can show the following theorem approximation of integrals suppose that th is twice continuously differentiable with unique global minimum at thand th is continuous with (th thenas ( ln (the- (thdth - (thln rp more generallyif rn has unique global minimum thn and rn thn ththen ln (the- rn (thdth - (thln rp proofwe only sketch the proof of ( let (thbe the hessian matrix of at th by taylor' theorem we can write (th (thr(th (th th (th th) (th)(th th)th { = we can exchange the limit and expectationas (th) {th th maxthth (thand |thmaxthth (thr th maxthth (thdth