to verify that the markov chain for the systematic gibbs sampler indeed has limiting pdf ( )we need to check that the global balance equations ( hold by integrating (in the continuous caseboth sides in ( with respect to xwe see that indeed (xq - ( xdx (yexample (gibbs sampler for the bayesian normal modelgibbs samplers are often applied in bayesian statisticsto sample from the posterior pdf consider for instance the bayesian normal model (us / ( us ( ihere the prior for (us is improper that isit is not pdf in itselfbut by obstinately applying bayesformula it does yield proper posterior pdf in some sense this prior conveys the least amount of information about and following the same procedure as in example we find the posterior pdfp - / - (xi ) ( (us xs exp improper prior note that and here are the "variablesand is fixed data vector to simulate samples and from ( using the gibbs samplerwe need the distributions of both ( xand ( uxto find ( )view the right-hand side of ( as function of onlyregarding as constant this gives ux nu xi exp ( xexp ( / ( ) exp ( / this shows that ( xhas normal distribution with mean and variance / similarlyto find ( ux)view the right-hand side of ( as function of regarding as constant this gives ( ux( )- / - exp ( / ( = showing that ( uxhas an inverse-gamma distribution with parameters / and pn = (xi / the gibbs sampler thus involves the repeated simulation of and ( uxinvgamma / (xi ) / ( xn xs / = simulating invgamma(alis achieved by first generating gamma(aland then returning /