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  "body": "\n## Summary\n\nDetailed balance is a property of a distribution $\\pi$ and a [[Markov chain|0G48PYM96Q]].\nIt says that, in equilibrium, the probability flow from each state $i$ to each state $j$ equals the flow from $j$ back to $i$.\nDetailed balance implies that $\\pi$ is a [[stationary distribution|0KRH5157WP]], and a chain with such a $\\pi$ is called reversible.\nIt is the usual design principle of Markov chain Monte Carlo algorithms.\n\n## Explanations\n\n### L1 human\n\nPicture many walkers spread over the states according to $\\pi$, all moving at once by the rules of the chain.\nStationarity only asks that the total number in each state stay the same.\nDetailed balance asks more: for every pair of states, as many walkers go from $i$ to $j$ as from $j$ to $i$.\nA film of the chain in equilibrium then looks the same when played backwards.\nA random walk on a graph that moves to a uniformly chosen neighbour has this property, with $\\pi$ proportional to the degrees.\n",
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