systematic gibbs sampler random-order gibbs sampler random gibbs sampler reversible gibbs sampler cycle consists of steps - in which the components are updated in fixed order for this reason algorithm is also called the systematic gibbs sampler in the random-order gibbs samplerthe order in which the components are updated in each cycle is random permutation of { (see exercise other modifications are to update the components in blocks ( several at the same time)or to update only random selection of components the variant where in each cycle only single random component is updated is called the random gibbs sampler in the reversible gibbs sampler cycle consists of the coordinate-wise updating in all casesexcept for the systematic gibbs samplerthe resulting markov chain {xt is reversible and hence its limiting distribution is precisely (xunfortunatelythe systematic gibbs markov chain is not reversible and so the detailed balance equations are not satisfied howevera similar result holdsdue to hammersley and cliffordunder the so-called positivity conditionif at point ( xn all marginal densities (xi nthen the joint density ( theorem hammersley-clifford balance for the gibbs sampler let - ( xdenote the transition density of the systematic gibbs samplerand let qn- ( ybe the transition density of the reverse movein the order thenif the positivity condition holdsf (xq - ( xf (yqn- ( ( prooffor the forward move we haveq - ( xf ( xn ( xn (yn yn- )and for the reverse moveqn- ( yf (xn yn- (xn- yn- xn ( xn consequentlyn (yi yi- xi+ xn - ( xqn- ( yf (xi yi- xi+ xn = ( yi xi+ xn ( yi- xi xn = ( yi xi+ xn (yn- qni= (xj= ( - xn (yn- (yi= ( yi xi+ xn qn- (xf (xj= ( + xn the result follows by rearranging the last identity the positivity condition ensures that we do not divide by along the line intuitivelythe long-run proportion of transitions for the "forward movechain is equal to the long-run proportion of transitions for the "reverse movechain