maximizing the log-likelihood ln ( bxwith respect to gives the maximum likelihood estimator of in supervised learning frameworkthis is equivalent to minimizingn ln (yi bxi ln ( bxn = =yi ln ( > ( yi ln( ( > ) = ( by comparing ( with ( )we see that we can interpret ( as the cross-entropy training loss associated with comparing true conditional pdf ( xwith an approximation pdf ( bxvia the loss function lossf ( ) ( bx):ln ( bx- ln (xb( yln( (xb)minimizing ( in terms of actually constitutes convex optimization problem since ln (xbln( - and ln( (xb)-xb ln( - )the cross-entropy training loss ( can be rewritten as  xh ( yi ) > ln -xi rt ( : = we leave it as exercise to show that the gradient rt (band hessian (bof rt (bare given by rt ( (ui yi xi ( = and (bui ( ui xi > = ( respectivelywhere ui : ( > bnotice that (bis positive semidefinite matrix for all values of bimplying the convexity of rt (bconsequentlywe can find an optimal efficientlye via newton' method specificallygiven an initial value for iteratively compute bt bt- - (bt- rt (bt- )( until the sequence is deemed to have convergedusing some pre-fixed convergence criterion figure shows the outcomes of independent bernoulli random variableswhere each success probability( exp(-( )))- depends on and - the true logistic curve is also shown (dashed linethe minimum training loss curve (red lineis obtained via the newton scheme ( )giving estimates bb - and bb the python code is given below