in other wordsthe maximum likelihood estimate th for th is obtained by solving the root of the average score functionthat isby solving st (th ( it is often not possible to find th in an explicit form in that case one needs to solve the equation ( numerically there exist many standard techniques for root-findinge via newton' method (see section )wherebystarting from an initial guess th subsequent iterates are obtained via the iterative scheme tht+ tht - (tht st (tht )where newton' method st (th (xi thht (th:th = th is the average hessian matrix of {ln (xi th)}ni= under (th)the expectation of ht (this equal to the information matrix (th)which does not depend on the data this suggests an alternative iterative schemecalled fisher' scoring methodtht+ tht - (tht st (tht )( fisher' scoring method which is not only easier to implement (if the information matrix can be readily evaluated)but also is more numerically stable example (maximum likelihood for the gamma distributionwe wish to approximate the density of the gamma(aldistribution for some true but unknown parameters aand lon the basis of training set { xn of iid samples from this distribution choosing our approximating function (alin the same class of gamma densitiesla xa- -lx ( ( alg(awith and we seek to solve ( taking the logarithm in ( )the loglikelihood function is given by ( al: ln ln ( ( ln lx it follows that ( alln ps(aln (ala - ( all where ps is the derivative of ln gthe so-called digamma function hence ( alal ( al-ps (aps (al (al- - la ( all( all al fisher' scoring method ( can now be used to solve ( )with ln ps(an- ni= ln xi st (ala - ni= xi and (alh(aldigamma function