Time-Reversible Markov Chains and Detailed Balance

6 TIME-REVERSIBLE MARKOV CHAINS AND DETAILED BALANCE 

So far, we have considered that we knew the one-step transition probabilities Pij for all values of i and j, and found the long-run distribution tt for all values of i from that. In this section we look at this the other way around. We start with the long-run distribution and want to find a Markov chain with that distribution. That means we find the set of one-step transition probabilities that will have that long run distribution. First we note that summing across rows 

Tlme-iW0rsible Msrkw chains. If we look at the states of a Markov chain in the reverse time order they also form a Markov chain called the backwards chain. Let the transition probabilities for the backwards chain be 

The Markov chain is said to be time reversible when the backwards Markov chain and the forward Markov chain have the same transition probabilities. In other words q J = p j for all states i and j. Then it follows that the transition probabilities satisfy 

Theorem 4 A set of transition probabilities satisfying the detailed balance condition will have steady state distribution n. 

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