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  "body": "\n## Summary\n\nThe posterior probability of a hypothesis $H$ is its [[conditional probability|0FJB06AD2P]] $P(H \\mid D)$ given the observed data $D$.\nBy [[Bayes' theorem|0NM6E9VW04]], $P(H \\mid D) = P(D \\mid H) \\, P(H) / P(D)$. So the posterior is proportional to the likelihood $P(D \\mid H)$ times the [[prior probability|00NBBSXRQS]] $P(H)$.\nFor an unknown parameter $\\theta$ with prior density $\\pi(\\theta)$ and likelihood $L(\\theta \\mid x)$, the posterior density is $\\pi(\\theta \\mid x) = L(\\theta \\mid x) \\, \\pi(\\theta) / \\int L(t \\mid x) \\, \\pi(t) \\, dt$.\n\n## Remarks\n\nThe posterior of one experiment can be the prior of the next experiment. So Bayesian updates can be done one observation at a time.\nA credible interval is an interval that has a given posterior probability, for example $95\\%$.\n\n## Explanations\n\n### L1 human\n\nThe posterior is what you believe after you see the data.\nIt mixes what you believed before (the prior) with how well each hypothesis explains the data (the likelihood).\n",
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