Question

For a time reversible Markov chain, argue that the rate at which transitions from \(i\) to \(j\) to \(k\) occur must equal the rate at which transitions from \(k\) to \(j\) to \(i\) occur.

Short answer

For a time-reversible Markov chain, the detailed balance condition states that \(\pi_i P_{ij} = \pi_j P_{ji}\). Using this property, we can express the transition rate from i to j to k as \(\pi_i P_{ij} P_{jk}\) and from k to j to i as \(\pi_k P_{kj} P_{ji}\). Applying detailed balance, we find that these two rates are equal, confirming the statement for the time-reversible Markov chain.

Step-by-step solution

Understanding Time Reversibility

In a Markov chain, time reversibility means that the process behaves identically when the time direction is reversed. Mathematically, this can be expressed using the concept of detailed balance: \(\pi_i P_{ij} = \pi_j P_{ji}\) where \(\pi_i\) represents the stationary distribution (i.e., the probability of being in state i in the long run), \(P_{ij}\) refers to the probability of transitioning from state i to state j, and \(P_{ji}\) refers to the probability of transitioning from state j to state i. Now we want to show that the rate of transitions from i to j to k is equal to the rate of transitions from k to j to i.

Transition Rate from i to j to k

First, let's calculate the rate of transition transitions from i to j to k. Since we're dealing with discrete states, the transition rate can be defined as a product of transition probabilities and the probabilities to be in the corresponding state: Rate(i to j to k) = \(\pi_i P_{ij} P_{jk}\)

Transition Rate from k to j to i

Similarly, let's calculate the rate of transition transitions from k to j to i: Rate(k to j to i) = \(\pi_k P_{kj} P_{ji}\)

Applying Time Reversibility

Using the detailed balance condition explained in Step 1, we can substitute transition probabilities in the rate expressions: Rate(i to j to k) = \(\pi_i \frac{\pi_j P_{ji}}{\pi_i} \frac{\pi_k P_{jk}}{\pi_j} = \pi_k P_{kj} P_{ji}\) Comparing the expressions for the transition rate from i to j to k and the transition rate from k to j to i: \(\pi_k P_{kj} P_{ji} = \pi_k P_{kj} P_{ji}\) which shows that the rate of transitions from i to j to k is indeed equal to the rate of transitions from k to j to i, confirming the statement for the time-reversible Markov chain.