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  "body": "\n## Summary\n\nA conjugate prior for a likelihood is a [[prior distribution|00NBBSXRQS]] from a family that the update by Bayes' theorem maps into itself.\nThe [[posterior distribution|0KJ7ZS8AJA]] then lies in the same family, and only its parameters change.\nThis gives closed-form posteriors, which made conjugate priors central in [[Bayesian inference|0JWRJX2RMP]] before simulation methods became common.\n\n## Explanations\n\n### L1 human\n\nA Bayesian update multiplies the prior by the likelihood.\nFor a well-chosen family of priors the product has the same shape as the prior, with new parameters.\nFor a coin with unknown chance $p$, a beta prior $\\mathrm{Beta}(a, b)$ plus $k$ heads and $m$ tails gives the posterior $\\mathrm{Beta}(a + k, b + m)$.\nSo the prior acts like $a$ earlier heads and $b$ earlier tails, and the data just add to these counts.\n",
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