because the weight parameter gn goes to as ph the improper prior ( also allows us to simplify the posterior marginal for ns / -( - )/ - - ( tg(us tdu ( exp which we recognize as the density corresponding to - gamma sn in addition to ( we can also use an improper prior for if we take the limit and in ( )then we also obtain the improper prior ( / (or equivalently ( / / in this casethe posterior marginal density for implies thatns kh - and the posterior marginal density for implies thatu xn tn- nn ( in generalderiving simple formula for the posterior density of th is either impossible or too tedious insteadthe monte carlo methods in can be used to simulate (approximatelyfrom the posterior for the purposes of inference and prediction one way in which distributional result such as ( can be useful is in the construction of credible interval for the parameter uthat isan interval such that the probability [ tis equal to for examplethe symmetric credible interval is sn sn gxn xn - - where is the -quantile of the tn- distribution note that the credible interval is not random object and that the parameter is interpreted as random variable with distribution this is unlike the case of classical confidence intervalswhere the parameter is nonrandombut the interval is (the outcome ofa random object as generalization of the bayesian credible interval we can define - credible regionwhich is any set satisfying [th tg(th tdth ( thr credible interval credible region