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  "body": "\n## Summary\n\nMarkov chain Monte Carlo (MCMC) is a family of methods to sample from a probability distribution $\\pi$.\nThey run a [[Markov chain|0G48PYM96Q]] whose [[stationary distribution|0KRH5157WP]] is $\\pi$.\nAverages of a function along the path of the chain then estimate the expectation of that function under $\\pi$.\nMCMC works even when $\\pi$ is known only up to a normalising constant, which makes it a basic tool of Bayesian statistics and statistical physics.\nThe [[Metropolis–Hastings algorithm|MKHF449E22]] and the Gibbs sampler are the main examples.\n\n## Explanations\n\n### L1 human\n\nSome distributions are too complicated to sample from directly, for example the law of all configurations of a large magnet.\nMCMC instead designs a random walk on the configurations that, in the long run, visits each one with the right frequency.\nOne lets the walk run, records what it sees, and averages.\nThe price is that successive samples are dependent, so more of them are needed than with independent draws.\n",
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