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  "body": "\n## Statement\n\nLet $(\\Omega, \\mathcal{F}, P)$ be a probability space and let $A, B \\in \\mathcal{F}$ with $P(A) \u003e 0$ and $P(B) \u003e 0$.\nThen\n$$P(A \\mid B) = \\frac{P(B \\mid A) \\, P(A)}{P(B)}.$$\n\n## Remarks\n\nLet $A_1, A_2, \\dots$ be a finite or countable partition of $\\Omega$ into events of positive probability. With the [[law of total probability|0E5G5FMT2R]], $P(A_j \\mid B) = \\dfrac{P(B \\mid A_j) \\, P(A_j)}{\\sum_i P(B \\mid A_i) \\, P(A_i)}$.\nIn Bayesian language, $P(A)$ is the [[prior probability|00NBBSXRQS]], $P(B \\mid A)$ is the likelihood and $P(A \\mid B)$ is the [[posterior probability|0KJ7ZS8AJA]].\nThe proof: both $P(A \\mid B) \\, P(B)$ and $P(B \\mid A) \\, P(A)$ are equal to $P(A \\cap B)$.\n\n## History\n\nThe theorem is named after Thomas Bayes. Pierre-Simon Laplace found it independently and developed it further.\n\n## Explanations\n\n### L1 human\n\nBayes' theorem turns \"the chance of the evidence given the cause\" into \"the chance of the cause given the evidence\".\nA medical test tells you how often sick people test positive. Bayes' theorem tells you how likely a person with a positive test is to be sick.\n",
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