markov chain monte carlo target burn-in period markov chain monte carlo (mcmcis monte carlo sampling technique for (approximatelygenerating samples from an arbitrary distribution -often referred to as the target distribution the basic idea is to run markov chain long enough such that its limiting distribution is close to the target distribution often such markov chain is constructed to be reversibleso that the detailed balance equations ( can be used depending on the starting position of the markov chainthe initial random variables in the markov chain may have distribution that is significantly different from the target (limitingdistribution the random variables that are generated during this burn-in period are often discarded the remaining random variables form an approximate and dependent sample from the target distribution in the next two sections we discuss two popular mcmc samplersthe metropolishastings sampler and the gibbs sampler proposal acceptance probability markov chain monte carlo metropolis-hastings sampler the metropolis-hastings sampler [ is similar to the acceptance-rejection method in that it simulates trial statewhich is then accepted or rejected according to some random mechanism specificallysuppose we wish to sample from target pdf ( )where takes values in some -dimensional set the aim is to construct markov chain {xt in such way that its limiting pdf is suppose the markov chain is in state at time transition of the markov chain from state is carried out in two phases first proposal state is drawn from transition density (xthis state is accepted as the new statewith acceptance probability (yq( , (xymin (xq( ( or rejected otherwise in the latter case the chain remains in state the algorithm just described can be summarized as follows algorithm metropolis-hastings sampler inputinitial state sample size ntarget pdf ( )proposal function ( xoutputx (dependent)approximately distributed according to ( for to do draw ( xt /draw proposal (xt /acceptance probability as in ( draw ( if then xt+ else xt+ xt return the fact that the limiting distribution of the metropolis-hastings markov chain is equal to the target distribution (under general conditionsis consequence of the following result