To carry out this Markov based sampling, we need now to describe how to obtain the two distributions P(X11=i,X12=j∣H,R) and P(X{s}1=d,X{s}2=f∣X{s−1}1=i,X{s−1}2=j,H,R). To do so, we decompose them by using equations (1) and (2) as follows: P(X{1}1,X{1}2∣H,R)=P(R∣X{1}1,X{1}2)P(X{1}1,X{1}2∣H) P(X{s}1,X{s}2∣X{s−1}1,X{s−1}2,H,R)∝P(X{s}1,X{s}2,X{s−1}1,X{s−1}2∣H,R)∝P(R∣X{s}1,X{s}2,X{s−1}1,X{s−1}2)P(X{s}1,X{s}2,X{s−1}1,X{s−1}2∣H)
