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  "body": "\n## Definition\n\nLet $P = (p_{ij})_{i, j \\in S}$ be the [[transition matrix|0F462493EA]] of a time-homogeneous [[Markov chain|033TJBZDG0]] on a countable set $S$.\nA probability distribution $\\pi = (\\pi_i)_{i \\in S}$ on $S$ satisfies *detailed balance* for $P$ if\n$$\\pi_i \\, p_{ij} = \\pi_j \\, p_{ji} \\quad \\text{for all } i, j \\in S.$$\nThe chain is then called *reversible* with respect to $\\pi$.\n\n## Remarks\n\nDetailed balance implies that $\\pi$ is [[stationary|0KRH5157WP]]: $\\sum_i \\pi_i p_{ij} = \\sum_i \\pi_j p_{ji} = \\pi_j$.\nIf $X_0$ has law $\\pi$, then $(X_0, \\ldots, X_n)$ and $(X_n, \\ldots, X_0)$ have the same law.\nExample: on a finite connected [[graph|00C0BR7D7H]] with edge set $E$, let $p_{ij} = 1/\\deg(i)$ when $j$ is a neighbour of $i$.\nWith $\\pi_i = \\deg(i) / (2\\abs{E})$, both sides equal $1/(2\\abs{E})$ for neighbours, and both vanish otherwise.\nNon-example: on $S = \\set{1, 2, 3}$ let the chain move deterministically $1 \\to 2 \\to 3 \\to 1$.\nThe uniform law is stationary, but $\\pi_1 p_{12} = 1/3$ while $\\pi_2 p_{21} = 0$, so detailed balance fails.\nKolmogorov'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.\n",
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