91Ó°ÊÓ

A time-reversible Markov chain is a Markov chain with the property that the probability of being in a given sequence of states does not change, regardless of whether the sequence is traversed forward or backward in time. In other words, for all states \(i\), \(j\), and \(k\), a Markov chain is time-reversible if and only if the probability of going from state \(i\) to \(j\) and then to \(k\) is equal to the probability of going from state \(k\) to \(j\) and then to \(i\).

A necessary and sufficient condition for a Markov chain to be time-reversible is that the detailed balance condition holds, which is given by \[ \pi(i)P_{ij} = \pi(j)P_{ji} \] for all states \(i\) and \(j\), where \(\pi(i)\) is the equilibrium distribution probability of state i and \(P_{ij}\) represents the transition probability from state \(i\) to state \(j\).

We want to show that the rate at which transitions occur from state \(i\) to \(j\) and then to \(k\) must be equal to the rate at which transitions occur from state \(k\) to \(j\) and then to \(i\). Let us denote the rates as: - Rate from \(i\) to \(j\) to \(k\): \(R_{i \to j \to k}\) - Rate from \(k\) to \(j\) to \(i\): \(R_{k \to j \to i}\) We can rewrite the given statement as: \[ R_{i \to j \to k} = R_{k \to j \to i} \] Now, let's compute \(R_{i \to j \to k}\) and \(R_{k \to j \to i}\) using the detailed balance condition. Using the detailed balance condition for states \(i\) and \(j\), and states \(j\) and \(k\), we have: \[ \pi(i)P_{ij} = \pi(j)P_{ji} \quad \text{and} \quad \pi(j)P_{jk} = \pi(k)P_{kj} \] We want to express \(R_{i \to j \to k}\) in terms of equilibrium distribution probabilities and transition probabilities. This can be done as follows: \[ R_{i \to j \to k} = \pi(i)P_{ij}P_{jk} \] Similarly, let's calculate the rate at which transitions occur from state \(k\) to \(j\) and then to \(i\): \[ R_{k \to j \to i} = \pi(k)P_{kj}P_{ji} \] Now, we can substitute the balanced equations from above into these expressions for \(R_{i \to j \to k}\) and \(R_{k \to j \to i}\) respectively: \[ R_{i \to j \to k} = (\pi(j)P_{ji})(\pi(k)P_{kj}) \quad \text{and} \quad R_{k \to j \to i} = (\pi(k)P_{kj})(\pi(j)P_{ji}) \] These expressions are equal, so our statement is proven true: \[ R_{i \to j \to k} = R_{k \to j \to i} \] So, for a time-reversible Markov chain, the rate at which transitions occur from state \(i\) to \(j\) and then to \(k\) must be equal to the rate at which transitions occur from state \(k\) to \(j\) and then to \(i\).