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Revision as of 23:05, 16 October 2016
Welcome to the explain xkcd wiki!
We have an explanation for all 2058 xkcd comics,
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Modified Bayes' Theorem 
Title text: Don't forget to add another term for "probability that the Modified Bayes' Theorem is correct." 
Explanation
This explanation may be incomplete or incorrect: When using the Mathsyntax please also care for a proper layout. Please edit the explanation below and only mention here why it isn't complete. Do NOT delete this tag too soon. 
Bayes' Theorem is an equation in statistics that gives the probability of a given hypothesis accounting not only for a single experiment or observation but also for your existing knowledge about the hypothesis, i.e. its prior probability. Randall's modified form of the equation also purports to account for the probability that you are indeed applying Bayes' Theorem itself correctly by including that as a term in the equation.
Bayes' theorem is:
, where
 is the probability that , the hypothesis, is true given observation . This is called the posterior probability.
 is the probability that observation will appear given the truth of hypothesis . This term is often called the likelihood.
 is the probability that hypothesis is true before any observations. This is called the prior, or belief.
 is the probability of the observation regardless of any hypothesis might have produced it. This term is called the marginal likelihood.
The purpose of Bayesian inference is to discover something we want to know (how likely is it that our explanation is correct given the evidence we've seen) by mathematically expressing it in terms of things we can find out: how likely are our observations, how likely is our hypothesis a priori, and how likely are we to see the observations we've seen assuming our hypothesis is true. A Bayesian learning system will iterate over available observations, each time using the likelihood of new observations to update its priors (beliefs) with the hope that, after seeing enough data points, the prior and posterior will converge to a single model.
If the modified theorem reverts to the original Bayes' theorem (which makes sense, as a probability one would mean certainty that you are using Bayes' theorem correctly).
If the modified theorem becomes , which says that the belief in your hypothesis is not affected by the result of the observation (which makes sense because you're certain you're misapplying the theorem so the outcome of the calculation shouldn't affect your belief.)
This happens because the modified theorem can be rewritten as: . This is the linearinterpolated weighted average of the belief you had before the calculation and the belief you would have if you applied the theorem correctly. This goes smoothly from not believing your calculation at all (keeping the same belief as before) if to changing your belief exactly as Bayes' theorem suggests if . (Note that is the probability that you are using the theorem incorrectly.)
The title text suggests that an additional term should be added for the probability that the Modified Bayes Theorem is correct. But that's this equation, so it would make the formula selfreferential, unless we call the result the Modified Modified Bayes Theorem (or Modified^{2}). It could also result in an infinite regress  we'd need another term for the probability that the version with the probability added is correct, and another term for that version, and so on. If the modifications have a limit, then we can make that the Modified^{ω} Bayes Theorem, but then we need another term for whether we did that correctly, leading to the Modified^{ω+1} Bayes Theorem, and so on through every ordinal number. It's also unclear what the point of using an equation we're not sure of is (although sometimes we can: Newton's Laws are not as correct as Einstein's Theory of Relativity but they're a reasonable approximation in most circumstances. Alternatively, ask any student taking a difficult exam with a formula sheet.).
If we denote the probability that the Modified^{n} Bayes' Theorem is correct by , then one way to define this sequence of modified Bayes' theorems is by the rule
One can then show by induction that
If we assume that one doubts each step infinitely (that is, for any ), then we can calculate a limit.
Without changing a limit, we can assume that where . Then we can write a limit as:
where both and , so
This may interpreted as an universal truth that we have to trust something eventually, otherwise everything boils down to an unconditional belief.
Transcript
This transcript is incomplete. Please help editing it! Thanks. 
 Modified Bayes' theorem:
 P(HX) = P(H) × (1 + P(C) × ( P(XH)/P(X)  1 ))
 H: Hypothesis
 X: Observation
 P(H): Prior probability that H is true
 P(X): Prior probability of observing X
 P(C): Probability that you're using Bayesian statistics correctly
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You can read a brief introduction about this wiki at explain xkcd. Feel free to sign up for an account and contribute to the wiki! We need explanations for comics, characters, themes and everything in between. If it is referenced in an xkcd web comic, it should be here.
 If you're new to wiki editing, see how to edit wiki pages. There's also a handy wikicode cheatsheet.
 Discussion about the wiki itself happens at the Community portal.
 You can browse the comics from List of all comics or by navigating the category tree at Category:Comics.
 There are incomplete explanations listed here. Feel free to help out by expanding them!
 We sell advertising space to pay for our server costs. To learn more, go here.
Rules
Don't be a jerk.
There are a lot of comics that don't have setinstone explanations; feel free to put multiple interpretations in the wiki page for each comic.
If you want to talk about a specific comic, use its discussion page.
Please only submit material directly related to (and helping everyone better understand) xkcd... and of course only submit material that can legally be posted (and freely edited). Offtopic or other inappropriate content is subject to removal or modification at admin discretion, and users who repeatedly post such content will be blocked.
If you need assistance from an admin, post a message to the Admin requests board.