- Choose an initial state at time .
- Suppose that at time . Then pick according to .
- Move to with the probability , or stay with the probability .
Transition probability. One should choose to be irreducible on , and call it a proposal chain. In terms of transition probability, the above algorithm is given by
Reversibility. The detailed balance clearly holds for either or . Let . Suppose that . Then and , and therefore, we obtain
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