Detailed balance

Remarks

Detailed balance implies that $\pi$ is stationary: $\sum_i \pi_i p_{ij} = \sum_i \pi_j p_{ji} = \pi_j$. If $X_0$ has law $\pi$, then $(X_0, \ldots, X_n)$ and $(X_n, \ldots, X_0)$ have the same law. Example: on a finite connected graph with edge set $E$, let $p_{ij} = 1/\deg(i)$ when $j$ is a neighbour of $i$. With $\pi_i = \deg(i) / (2\abs{E})$, both sides equal $1/(2\abs{E})$ for neighbours, and both vanish otherwise. Non-example: on $S = \set{1, 2, 3}$ let the chain move deterministically $1 \to 2 \to 3 \to 1$. The uniform law is stationary, but $\pi_1 p_{12} = 1/3$ while $\pi_2 p_{21} = 0$, so detailed balance fails. Kolmogorov's criterion says that an irreducible chain with a stationary distribution is reversible if and only if $p_{i_1 i_2} p_{i_2 i_3} \cdots p_{i_n i_1} = p_{i_1 i_n} p_{i_n i_{n-1}} \cdots p_{i_2 i_1}$ for every finite cycle of states.