▁Monte ▁Carlo ▁sampling ▁can ▁only ▁approximate ▁the ▁target ▁distribution , ▁as ▁there ▁is ▁always ▁some ▁resid ual ▁effect ▁of ▁the ▁starting ▁position . ▁More ▁s oph istic ated ▁Mark ov ▁chain ▁Monte ▁Carlo - based ▁algorithms ▁such ▁as ▁coupling ▁from ▁the ▁past ▁can ▁produce ▁exact ▁samples , ▁at ▁the ▁cost ▁of ▁additional ▁computation ▁and ▁an ▁un bounded ▁( though ▁finite ▁in ▁expectation ) ▁running ▁time . ▁ ▁Many ▁random ▁walk ▁Monte ▁Carlo ▁methods ▁move ▁around ▁the ▁equilibrium ▁distribution ▁in ▁relatively ▁small ▁steps , ▁with ▁no ▁t endency ▁for ▁the ▁steps ▁to ▁proceed ▁in ▁the ▁same ▁direction . ▁These ▁methods ▁are ▁easy ▁to ▁implement ▁and ▁analyze , ▁but ▁unfortunately ▁it ▁can ▁take ▁a ▁long ▁time ▁for ▁the ▁walk er ▁to ▁explore ▁all ▁of ▁the ▁space . ▁The ▁walk er ▁will ▁often ▁double ▁back ▁and ▁cover ▁ground ▁already ▁covered . ▁ ▁Further ▁consideration ▁of ▁convergence ▁is ▁at ▁Mark ov ▁chain ▁central ▁limit ▁theorem . ▁See ▁ ▁for ▁a ▁discussion ▁of ▁the ▁theory ▁related ▁to ▁convergence ▁and ▁station ar ity ▁of ▁the ▁Metropol is - H ast ings ▁algorithm . ▁ ▁Software ▁ ▁Several ▁software ▁programs ▁provide ▁MC MC ▁sampling ▁capabilities , ▁for ▁example : ▁Pack ages ▁that ▁use ▁dialect s ▁of ▁the ▁B U GS ▁model ▁language : ▁Win BU GS ▁/ ▁Open BU GS / ▁Multi BU GS ▁J AG S ▁N IM B LE ▁gre ta , ▁a ▁Bay esian ▁statistical ▁model ing ▁language ▁/ ▁R ▁package ▁which ▁uses ▁T ensor Flow ▁behind ▁the ▁scenes , ▁similar ▁to ▁Py MC 3 ' s