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  "body": "\n## Summary\n\nBayesian inference is a method of statistical inference that treats an unknown parameter $\\theta$ as a random variable. It uses [[Bayes' theorem|0NM6E9VW04]] to update a probability distribution for $\\theta$ when data arrive.\nOne starts from a [[prior distribution|00NBBSXRQS]] with density $\\pi(\\theta)$ and a model with [[likelihood|05FSTXM92W]] $L(\\theta \\mid x) = f(x \\mid \\theta)$. The [[posterior distribution|0KJ7ZS8AJA]] has the density\n$$\\pi(\\theta \\mid x) = \\frac{f(x \\mid \\theta)\\, \\pi(\\theta)}{\\int_\\Theta f(x \\mid \\theta')\\, \\pi(\\theta')\\, d\\theta'}.$$\n\n## Remarks\n\nIn short, the posterior is proportional to the likelihood times the prior.\nExample: with a $\\operatorname{Beta}(a, b)$ prior for the success probability $p$, and $k$ successes in $n$ Bernoulli($p$) trials, the posterior is $\\operatorname{Beta}(a + k, b + n - k)$ (see [[beta distribution|0PP5ZAJPP7]]). The prior is conjugate.\nThe posterior gives point estimates, such as the posterior mean, and credible intervals.\nFor new observations, the posterior predictive distribution is $f(\\tilde{x} \\mid x) = \\int_\\Theta f(\\tilde{x} \\mid \\theta)\\, \\pi(\\theta \\mid x)\\, d\\theta$, if $\\tilde{x}$ and $x$ are conditionally independent given $\\theta$.\n\n## Explanations\n\n### L1 human\n\nBayesian inference starts with a belief about an unknown quantity, written as a probability distribution.\nEach new piece of data updates this belief with Bayes' theorem. The result, the posterior, combines the prior belief with the evidence.\n",
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