Detailed Balance

Consider a Markov chain where on the long run, for all states \(i\) and \(j\), the proportion (or chance) of states moving from \(i\) to \(j\) is the same as from \(j\) to \(i\). This suggest the following relation,

\[ \pi(i) P(i,j) = \pi(j) P(j,i) \quad \forall i, j \in S \]

We call this Markov chain to be detailed balanced and it turns out this satisfies the balance equation,

\[\begin{split} \begin{align*} \pi(j) &= \sum_{i \in S}\pi(i)P(i,j)\\ &= \sum_{i \in S}\pi(j)P(j,i)\\ &= \pi(j)\sum_{i \in S}P(j,i) \\ &= \pi(j) \end{align*} \end{split}\]