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  "body": "\n## Definition\n\nLet $S$ be a countable set and let $\\pi$ be a probability distribution on $S$ with $\\pi(x) \u003e 0$ for all $x \\in S$.\nLet $Q = (Q(x, y))_{x, y \\in S}$ be a [[transition matrix|0F462493EA]], the *proposal*, such that $Q(x, y) \u003e 0$ if and only if $Q(y, x) \u003e 0$.\nFor $x \\ne y$ with $Q(x, y) \u003e 0$, the *acceptance probability* is\n$$\\alpha(x, y) = \\min\\left(1, \\frac{\\pi(y) \\, Q(y, x)}{\\pi(x) \\, Q(x, y)}\\right).$$\nThe *Metropolis–Hastings chain* is the [[Markov chain|033TJBZDG0]] $(X_n)$ built as follows.\nGiven $X_n = x$, draw $Y$ from $Q(x, \\cdot)$.\nWith probability $\\alpha(x, Y)$ set $X_{n+1} = Y$; otherwise set $X_{n+1} = x$.\nIts transition matrix has $P(x, y) = Q(x, y) \\, \\alpha(x, y)$ for $y \\ne x$ and $P(x, x) = 1 - \\sum_{y \\ne x} P(x, y)$.\n\n## Remarks\n\nFor $x \\ne y$, $\\pi(x) P(x, y) = \\min\\left(\\pi(x) Q(x, y), \\pi(y) Q(y, x)\\right)$, which is symmetric in $x$ and $y$.\nSo $P$ satisfies [[detailed balance|CTGN6CK1ZS]] with respect to $\\pi$, and $\\pi$ is a [[stationary distribution|0KRH5157WP]].\nOnly the ratio $\\pi(y)/\\pi(x)$ is used, so $\\pi$ may be known up to a constant factor.\nExample: if $Q$ is symmetric, $\\alpha(x, y) = \\min(1, \\pi(y)/\\pi(x))$; this special case is the Metropolis algorithm.\nNon-example: the chain that always accepts the proposal has transition matrix $Q$, whose stationary law is in general not $\\pi$; it skips the acceptance step.\nThe chain is a [[Markov chain Monte Carlo|ZGNBR2SPZ0]] method when it is also irreducible.\nThe same construction works on general state spaces with densities in place of $\\pi$ and $Q$.\n",
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