example (bayesian regularization of maximum likelihoodconsider modeling the number of deaths during birth in maternity ward suppose that the hospital data consists of { xn }with xi if the -th baby has died during birth and xi otherwisefor possible bayesian model for the data is th ( (uniform iid priorwith ( xn thber(ththe likelihood is therefore ( thn = th xi ( th) -xi th ( th) - where xn is the total number of deaths since (th the posterior pdf is (th tth ( th) - th [ ]which is the pdf of the beta( distribution the normalization constant is ( ns the posterior pdf is shown in figure for (sn( it is not difficult figure posterior pdf for thwith and maximum posteriori to see that the maximum posteriori (mapestimate of th (the mode or maximizer of the posterior densityis argmax (th tn th which agrees with the maximum likelihood estimate figure also shows that the left one-sided credible interval for th is [ ]where is the quantile (roundedof the beta( distribution observe that when (sn( the maximum likelihood estimate th infers that deaths at birth are not possible we know that this inference is wrong -the probability of death can never be zeroit is simply (and fortunatelytoo small to be inferred accurately from sample size of in contrast to the maximum likelihood estimatethe posterior mean [th ( )/( is not zero for (sn( and provides the more reasonable point estimate of for the probability of death