lambda - *pi lambda xx pi lambda exp(-sqrt( ** ** / *sin ( sqrt( ** ** ))+ )* ( )* ( )* ( )* ( xx np zeros (( , ) np zeros (( , )for in range ( , ) randn ( , alpha np amin (( ( [ ][ , [ ][ ]/ ( [ ][ , [ ][ ], ) rand (alpha * ( - )* xx[ ,: plt scatter (xx [, xx [, alpha = , = plt axis('equal 'plt show ( gibbs sampler the gibbs sampler [ uses somewhat different methodology from the metropolishastings algorithm and is particularly useful for generating -dimensional random vectors the key idea of the gibbs sampler is to update the components of the random vector one at timeby sampling them from conditional pdfs thusgibbs sampling can be advantageous if it is easier to sample from the conditional distributions than from the joint distribution specificallysuppose that we wish to sample random vector [ xn ]according to target pdf (xlet (xi xi- xi+ xn represent the conditional pdf of the -th componentxi given the other components xi- xi+ xn the gibbs sampling algorithm is as follows gibbs sampler algorithm gibbs sampler inputinitial point sample size nand target pdf outputx approximately distributed according to for to do draw from the conditional pdf ( xt, xt, for to do draw yi from the conditional pdf (yi yi- xt, + xt, xt+ return there exist many variants of the gibbs samplerdepending on the steps required to update xt to xt+ -called the cycle of the gibbs algorithm in the algorithm abovethe in this section we employ bayesian notation styleusing the same letter for different (conditionaldensities cycle