theorem local balance for the metropolis-hastings sampler the transition density of the metropolis-hastings markov chain satisfies the detailed balance equations proofwe prove the theorem for the discrete case only because transition of the metropolis-hastings markov chain consists of two stepsthe one-step transition probability to go from to is not ( xbut if xq( xa(xy) ( ( , ( xa(xz)if we thus need to show that (xe ( xf (ye ( yfor all xy ( with the acceptance probability as in ( )we need to check ( for three cases(ax (bx and ( ) ( ( ) ( )and (cx and ( ) ( yf ( ) ( xcase (aholds trivially for case ( ) (xyf ( ) ( )/ ( ) ( )and (yx consequentlye ( xf ( ) ( ) (xand ( yq( )so that ( holds similarlyfor case (cwe have (xy and (yxf ( ) ( ) ( ) ( )it follows thate ( xq( xso that ( holds again and ( yf ( ) ( ) ( )thus if the metropolis-hastings markov chain is ergodicthen its limiting pdf is (xa fortunate property of the algorithmwhich is important in many applicationsis that in order to evaluate the acceptance probability (xyin ( )one only needs to know the target pdf (xup to constantthat is (xc (xfor some known function (xbut unknown constant the efficiency of the algorithm depends of course on the choice of the proposal transition density ( xideallywe would like ( xto be "closeto the target ( )irrespective of we discuss two common approaches choose the proposal transition density ( xindependent of xthat isq( xg(yfor some pdf (yan mcmc sampler of this type is called an independence sampler the acceptance probability is thus (yg(xa(xymin , (xg( independence sampler