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  "body": "\n## Definition\n\nLet $S$ be a countable set, let $\\pi$ be a probability distribution on $S$, and let $f \\colon S \\to \\R$ be bounded.\nA *Markov chain Monte Carlo method* for estimating $\\pi(f) = \\sum_{x \\in S} f(x) \\pi(x)$ consists of:\n\n1. a [[transition matrix|0F462493EA]] $P$ on $S$ such that $\\pi$ is a [[stationary distribution|0KRH5157WP]] of $P$ and the chain with transition matrix $P$ is irreducible;\n2. a [[Markov chain|033TJBZDG0]] $(X_k)_{k \\ge 0}$ with transition matrix $P$, simulated from some starting state;\n3. the estimator $\\hat{\\pi}_n(f) = \\frac{1}{n} \\sum_{k=1}^{n} f(X_k)$, computed for a large $n$.\n\n## Remarks\n\nBy the ergodic theorem for irreducible chains with a stationary distribution, $\\hat{\\pi}_n(f) \\to \\pi(f)$ almost surely as $n \\to \\infty$, for every starting state.\nThe usual way to obtain clause 1 is to impose [[detailed balance|CTGN6CK1ZS]] for $\\pi$, as the [[Metropolis–Hastings algorithm|MKHF449E22]] does.\nIn practice the first steps, the burn-in, are often discarded, and if the chain is also aperiodic then the law of $X_k$ tends to $\\pi$ by the [[convergence theorem|B1E1ES1BYD]].\nExample: on $S = \\Z$ with $\\pi(x) \\propto e^{-x^2}$, propose $x \\pm 1$ with probability $1/2$ each and accept with probability $\\min(1, \\pi(y)/\\pi(x))$.\nThis chain is irreducible and satisfies detailed balance for $\\pi$.\nNon-example: the simple [[random walk|04VZE3FZXF]] on the path $\\set{1, \\ldots, m}$, which moves to a uniform neighbour, has stationary law proportional to the degrees.\nSo its averages estimate expectations under that law, not under the uniform law, and it fails clause 1 for uniform $\\pi$ when $m \\ge 3$.\n",
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