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text: '[summary: [1lz Bayes' rule] generalizes to continuous [3jy functions], and says, "The [1rp posterior] probability [density](https://en.wikipedia.org/wiki/Probability_density_function) is *proportional* to the [1rq likelihood] function times the [1rm prior] probability [density](https://en.wikipedia.org/wiki/Probability_density_function)."\n\n$$\\mathbb P(H_x\\mid e) \\propto \\mathcal L_e(H_x) \\cdot \\mathbb P(H_x)$$]\n\n[1lz Bayes' rule] generalizes to continuous [3jy functions], and states, "The [1rp posterior] probability [density](https://en.wikipedia.org/wiki/Probability_density_function) is *proportional* to the [1rq likelihood] function times the [1rm prior] probability [density](https://en.wikipedia.org/wiki/Probability_density_function)."\n\n$$\\mathbb P(H_x\\mid e) \\propto \\mathcal L_e(H_x) \\cdot \\mathbb P(H_x)$$\n\n## Example\n\nSuppose we have a biased coin with an unknown bias $b$ between 0 and 1 of coming up heads on each individual coinflip. Since the bias $b$ is a continuous variable, we express our beliefs about the coin's bias using a [probability density function](https://en.wikipedia.org/wiki/Probability_density_function) $\\mathbb P(b),$ where $\\mathbb P(b)\\cdot \\mathrm{d}b$ is the probability that $b$ is in the interval $[b + \\mathrm{d}b]$ for $\\mathrm db$ small. (Specifically, the probability that $b$ is in the interval $[a, b]$ is $\\int_a^b \\mathbb P(b) \\, \\mathrm db.$)\n\nBy hypothesis, we start out completely ignorant of the bias $b,$ meaning that all initial values for $b$ are equally likely. Thus, $\\mathbb P(b) = 1$ for all values of $b,$ which means that $\\mathbb P(b)\\, \\mathrm db = \\mathrm db$ (e.g., the chance of $b$ being found in the interval from 0.72 to 0.76 is 0.04).\n\n\n\nWe then flip the coin, and observe it to come up tails. This is our first piece of evidence. The likelihood $\\mathcal L_{t_1}(b)$ of observation $t_1$ given bias $b$ is a continuous function of $b$, equal to 0.4 if $b = 0.6,$ 0.67 if $b = 0.33,$ and so on (because $b$ is the probability of heads and the observation was tails).\n\nGraphing the likelihood function $\\mathcal L_{t_1}(b)$ as it takes in the fixed evidence $t_1$ and ranges over variable $b,$ we obtain the straightforward graph $\\mathcal L_{t_1}(b) = 1 - b.$\n\n\n\nIf we multiply the likelihood function by the prior probability function as it ranges over $b$, we obtain a *relative probability* function on the posterior, $\\mathbb O(b\\mid t_1) = \\mathcal L_{t_1}(b) \\cdot \\mathbb P(b) = 1 - b,$ which gives us the same graph again:\n\n\n\nBut this can't be our posterior *probability* function because it doesn't integrate to 1. $\\int_0^1 (1 - b) \\, \\mathrm db = \\frac{1}{2}.$ (The area under a triangle is half the base times the height.) [1rk Normalizing] this relative probability function will give us the posterior probability function:\n\n$\\mathbb P(b \\mid t_1) = \\dfrac{\\mathbb O(b \\mid t_1)}{\\int_0^1 \\mathbb O(b \\mid t_1) \\, \\mathrm db} = 2 \\cdot (1 - f)$\n\n\n\nThe shapes are the same, and only the *y*-axis labels have changed to reflect the different heights of the pre-normalized and normalized function.[todo: Regraph these graphs with actual height changes]\n\nSuppose we now flip the coin another two times, and it comes up heads then tails. We'll denote this piece of evidence $h_2t_3.$ Although these two coin tosses pull our beliefs about $b$ in opposite directions, they don't cancel out — far from it! In fact, one value of $b$ ("the coin is always tails") is completely eliminated by this evidence, and many extreme values of $b$ ("almost always heads" and "almost always tails") are hit badly. That is, while the heads and the coins tails pull our beliefs in opposite directions, they don't pull with the same strength on all possible values of $b.$\n\nWe multiply the old belief\n\n\n\nby the additional pieces of evidence\n\n\n\nand\n\n\n\nand obtain the posterior *relative* density\n\n\n\nwhich is proportional to the [1rk normalized] posterior probability\n\n\n\nWriting out the whole operation from scratch:\n\n$$\\mathbb P(b \\mid t_1h_2t_3) = \\frac{\\mathcal L_{t_1h_2t_3}(b) \\cdot \\mathbb P(b)}{\\mathbb P(t_1h_2t_3)} = \\frac{(1 - b) \\cdot b \\cdot (1 - b) \\cdot 1}{\\int_0^1 (1 - b) \\cdot b \\cdot (1 - b) \\cdot 1 \\, \\mathrm{d}b} = {12\\cdot b(1 - b)^2}$$\n\nNote that it's okay for a posterior probability density to be greater than 1, so long as the total probability *mass* isn't greater than 1. If there's probability density 1.2 over an interval of 0.1, that's only a probability of 0.12 for the true value to be found in that interval.\n\nThus, intuitively, Bayes' rule "just works" when calculating the posterior probability density from the prior probability density function and the (continuous) likelihood ratio function. A proof is beyond the scope of this guide; refer to [ Proof of Bayes' rule in the continuous case].',
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Summary: '[1lz Bayes' rule] generalizes to continuous [3jy functions], and says, "The [1rp posterior] probability [density](https://en.wikipedia.org/wiki/Probability_density_function) is *proportional* to the [1rq likelihood] function times the [1rm prior] probability [density](https://en.wikipedia.org/wiki/Probability_density_function)."\n\n$$\\mathbb P(H_x\\mid e) \\propto \\mathcal L_e(H_x) \\cdot \\mathbb P(H_x)$$'
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