Reversible Process

A Markov chain is considered a reversible chain if its reverse sequence \(Y_0, Y_1, \ldots, Y_n\) has the same one-step transition probability as the forward sequence \(X_0, X_1, \ldots, X_n\). This implies, a chain is reversible if the chain is in detailed balance with positive solution \(\pi(i)\) for all states \(i\),

\[ \pi(i) P(i,j) = \pi(j)P(j,i) \]

Proof

Given a sequence, the probability of the next step in the reverse sequence can be related to the foward sequence

\[ P(Y_{m+1} = j \mid Y_{m < i}, Y_i = i) = \frac{\pi(j)}{\pi(i)}P(j,i) \]

Since the RHS does not depend on the intermediate states \(m < i\),

\[ P(Y_{m+1} = j \mid Y_i = i) = \frac{\pi(j)}{\pi(i)}P(j,i) \]

Thus, the Markov chain is reversible if the probability of the foward step \(P(i, j)\) is the same as the probability of stepping backward from \(i\) back to \(j\) or \(P(Y_{m+1} = j \mid Y_i = i)\),

\[\begin{split} \begin{gather*} P(Y_{m+1} = j \mid Y_i = i) = P(i,j)\\ \Downarrow\\ \frac{\pi(j)}{\pi(i)}P(j,i) = P(i,j) \end{gather*} \end{split}\]

This is exactly the balanced equation,

\[ \pi(i) P(i,j) = \pi(j)P(j,i) \]

Birth and Death Chain

A birth and death is one that can only go to 3 states: up \(i+1\), down \(i-1\), and stay \(i\).

This simple process makes it not surprising that all birth and death chain are all reversible processes. As we can abritrarily swap out \(i\) and \(k\) and the process is reversed.