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text: 'Bayes' rule is the mathematics of [1rf probability theory] governing how to update your beliefs in the light of new evidence.\n\n[toc:]\n\n## [1y9 Notation]\n\nIn much of what follows, we'll use the following [1y9 notation]:\n\n- Let the hypotheses being considered be $H_1$ and $H_2$.\n- Let the evidence observed be $e_0.$\n- Let $\\mathbb P(H_i)$ denote the [1rm prior probability] of $H_i$ before observing the evidence.\n- Let the [1rj conditional probability] $\\mathbb P(e_0\\mid H_i)$ denote the [1rq likelihood] of observing evidence $e_0$ assuming $H_i$ to be true.\n- Let the [1rj conditional probability] $\\mathbb P(H_i\\mid e_0)$ denote the [1rp posterior probability] of $H_i$ after observing $e_0.$\n\n## [1x5 Odds]/[1zm proportional] form\n\nBayes' rule in the [1x5 odds form] or [1zm proportional form] states:\n\n$$\\dfrac{\\mathbb P(H_1)}{\\mathbb P(H_2)} \\times \\dfrac{\\mathbb P(e_0\\mid H_1)}{\\mathbb P(e_0\\mid H_2)} = \\dfrac{\\mathbb P(H_1\\mid e_0)}{\\mathbb P(H_2\\mid e_0)}$$\n\nIn other words, the [1rm prior] [1rb odds] times the [1rq likelihood ratio] yield the [1rp posterior] odds. [1rk Normalizing] these odds will then yield the posterior probabilities.\n\nIn [1zm other other words]: If you initially think $h_i$ is $\\alpha$ times as probable as $h_k$, and then see evidence that you're $\\beta$ times as likely to see if $h_i$ is true as if $h_k$ is true, you should update to thinking that $h_i$ is $\\alpha \\cdot \\beta$ times as probable as $h_k.$\n\nSuppose that Professor Plum and Miss Scarlet are two suspects in a murder, and that we start out thinking that Professor Plum is twice as likely to have committed the murder as Miss Scarlet ([1rm prior] [1rb odds] of 2 : 1). We then discover that the victim was poisoned. We think that Professor Plum is around one-fourth as likely to use poison as Miss Scarlet ([1rq likelihood ratio] of 1 : 4). Then after observing the victim was poisoned, we should think Plum is around half as likely to have committed the murder as Scarlet: $2 \\times \\dfrac{1}{4} = \\dfrac{1}{2}.$ This reflects [1rp posterior] odds of 1 : 2, or a posterior probability of 1/3, that Professor Plum did the deed.\n\n## [1xr Proof]\n\nThe [1xr proof of Bayes' rule] is by the definition of [1rj conditional probability] $\\mathbb P(X\\wedge Y) = \\mathbb P(X\\mid Y) \\cdot \\mathbb P(Y):$\n\n$$\n\\dfrac{\\mathbb P(H_i)}{\\mathbb P(H_j)} \\times \\dfrac{\\mathbb P(e\\mid H_i)}{\\mathbb P(e\\mid H_j)}\n= \\dfrac{\\mathbb P(e \\wedge H_i)}{\\mathbb P(e \\wedge H_j)}\n= \\dfrac{\\mathbb P(e \\wedge H_i) / \\mathbb P(e)}{\\mathbb P(e \\wedge H_j) / \\mathbb P(e)}\n= \\dfrac{\\mathbb P(H_i\\mid e)}{\\mathbb P(H_j\\mid e)}\n$$\n\n## [1zh Log odds form]\n\nThe [1zh log odds form of Bayes' rule] states:\n\n$$\\log \\left ( \\dfrac\n {\\mathbb P(H_i)}\n {\\mathbb P(H_j)}\n\\right )\n+\n\\log \\left ( \\dfrac\n {\\mathbb P(e\\mid H_i)}\n {\\mathbb P(e\\mid H_j)}\n\\right ) \n =\n\\log \\left ( \\dfrac\n {\\mathbb P(H_i\\mid e)}\n {\\mathbb P(H_j\\mid e)}\n\\right )\n$$\n\nE.g.: "A study of Chinese blood donors found that roughly 1 in 100,000 of them had HIV (as determined by a very reliable gold-standard test). The non-gold-standard test used for initial screening had a sensitivity of 99.7% and a specificity of 99.8%, meaning that it was 500 times as likely to return positive for infected as non-infected patients." Then our prior belief is -5 orders of magnitude against HIV, and if we then observe a positive test result, this is evidence of strength +2.7 orders of magnitude for HIV. Our posterior belief is -2.3 orders of magnitude, or odds of less than 1 to a 100, against HIV.\n\nIn log odds form, the same [22x strength of evidence] (log [1rq likelihood ratio]) always [1zh moves us the same additive distance] along a line representing strength of belief (also in log odds). If we measured distance in probabilities, then the same 2 : 1 likelihood ratio might move us a different distance along the probability line depending on whether we started with prior 10% probability or 50% probability.\n\n## Visualizations\n\nGraphical of visualizing Bayes' rule include [1wy frequency diagrams, the waterfall visualization], the [1zm spotlight visualization], the [1zh magnet visualization], and the [1xr Venn diagram for the proof].\n\n## Examples\n\nExamples of Bayes' rule may be found [1wt here].\n\n## [1zg Multiple hypotheses and updates]\n\nThe [1x5 odds form of Bayes' rule] works for odds ratios between more than two hypotheses, and applying multiple pieces of evidence. Suppose there's a bathtub full of coins. 1/2 of the coins are "fair" and have a 50% probability of producing heads on each coinflip; 1/3 of the coins produce 25% heads; and 1/6 produce 75% heads. You pull out a coin at random, flip it 3 times, and get the result HTH. You may legitimately calculate:\n\n$$\\begin{array}{rll}\n(1/2 : 1/3 : 1/6) \\cong & (3 : 2 : 1) & \\\\\n\\times & (2 : 1 : 3) & \\\\\n\\times & (2 : 3 : 1) & \\\\\n\\times & (2 : 1 : 3) & \\\\\n= & (24 : 6 : 9) & \\cong (8 : 2 : 3)\n\\end{array}$$\n\nSince multiple pieces of evidence may not be [conditional_independence conditionally independent] from one another, it is important to be aware of the [naive_bayes_assumption Naive Bayes assumption] and whether you are making it.\n\n## [554 Probability form]\n\nAs a formula for a single probability $\\mathbb P(H_i\\mid e),$ Bayes' rule states:\n\n$$\\mathbb P(H_i\\mid e) = \\dfrac{\\mathbb P(e\\mid H_i) \\cdot \\mathbb P(H_i)}{\\sum_k \\mathbb P(e\\mid H_k) \\cdot \\mathbb P(H_k)}$$\n\n## [1zj Functional form]\n\nIn [1zj functional form], Bayes' rule states:\n\n$$\\mathbb P(\\mathbf{H}\\mid e) \\propto \\mathbb P(e\\mid \\mathbf{H}) \\cdot \\mathbb P(\\mathbf{H}).$$\n\nThe posterior probability function over hypotheses given the evidence, is *proportional* to the likelihood function from the evidence to those hypotheses, times the prior probability function over those hypotheses.\n\nSince posterior probabilities over [1rd mutually exclusive and exhaustive] possibilities must sum to $1,$ [1rk normalizing] the product of the likelihood function and prior probability function will yield the exact posterior probability function.\n',
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