Editing 2545: Bayes' Theorem

{{comic
| number    = 2545
| date      = November 22, 2021
| title     = Bayes' Theorem
| image     = bayes_theorem.png
| titletext = <nowiki>P((B|A)|(A|B)) represents the probability that you'll mix up the order of the terms when using Bayesian notation.</nowiki>
}}

==Explanation==
{{w|Bayes' theorem}} describes the probability of an event, given knowledge of conditions related to the event. It is typically used to update the probability that a starting condition occurred, given an outcome. This can reveal unintuitive results when the probability involved is very small. For example, when testing a large number of people for a rare disease, even a fairly accurate test will produce more false positives than the number of people actually afflicted with the disease, and hence a positive result is more likely to be false than true.
{| class="wikitable"
|| || Tested positive || Tested negative || Total
|-
|| Affected || 0.1% || 0.0% || 0.1%
|-
|| Unaffected || 0.9% || 99% || 99.9%
|-
|| Total || 1% || 99% || 100%
|}
For example, if a test has a 100% sensitivity (first line, all those affected receive a positive result) and a 99% specificity (second line, 1% of the unaffected also receive a positive result), the interpretation of a positive test depends on the prevalence of the disease in the population. In the example case, the prevalence is 0.1% (third column), so that when the test result is positive (1% of the tests, left column) the subject is actually unaffected nine time out of ten. Although this would be a very performant test, given the relative prevalences involved it will produce overwhelmingly false positives among all positive results. (But, in this example, all those told they are not in danger &mdash; almost a hundred times more individuals than test positive &mdash; are correctly notified.)

For this same example, the Bayesian formula gives : 
::P( Affected | Positive ) = P( Positive | Affected ) * P( Affected ) / P( Positive ) = 100% * 0.1% / 1% = 10% 
::and P( Unaffected | Positive ) = P( Positive | Unaffected ) * P( Unaffected ) / P( Positive ) = 0.9009% * 99.9% / 1% = 90% 

In this comic, a teacher is presenting a problem which the students are supposed to use Bayes' theorem to solve. However, the off-panel student knows that they are studying Bayes' theorem, so they use that prior knowledge to guess the usual answer to such problems. The punch line is the caption - The student doesn't need to do the calculation because they're familiar with questions involving Bayes' theorem and how they often present the counterintuitive result to illustrate the importance of prevalence to the calculation.

There is perhaps also a self-referential situation here where the student has updated their prior probabilities a number of times for whether the answer was "Yes" to a question involving Bayes' Theorem. If their method of answering "Yes" to every such question has succeeded every time before then by Bayes' theorem they will have a lot of justification to continue to do until they start getting it wrong. The prevalence of Bayes Theorem questions that require the answer "No" might be small enough that this doesn't happen in any small number of times and so they learn nothing of the false-positive rate until that point in time. This could be interpreted as a criticism of Bayesian Statistics which may treat a judgement as well justified (e.g. getting the question right) despite lacking a clear understanding of mechanism (e.g. basing your answer to the question on the numbers provided). 

The title text refers to the mathematical definition of Bayes' theorem: P(A | B) = P(B|A) * P(A) / P(B). Here, P(A|B) represents the probability of some event A occurring, given that B has occurred. This is often referred to as "the probability of A given B". It can be hard to remember if P(A|B) means probability of A given B, or if it's B given A, especially when talking about the probability of an earlier cause given a later effect. Randall's joke is based on this difficulty. Here P((B|A)|(A|B)) is meant to be read as the probability that you ''write'' (B|A) given that the correct expression is (A|B), which makes it the probability that you got the order of the notation mixed up.

==Transcript==
:[Miss Lenhart is pointing with a pointer, held in her right hand, to a white-board with tables, what looks like formulae and lots of other unreadable text. She looks toward her off-panel class, from where a voice replies to her question.]
:Miss Lenhart: Given these prevalences, is it likely that the test result is a false positive?
:Off-panel voice: Well, this chapter is on Bayes' Theorem, so yes.

:[Caption below the panel]:
:Sometimes, if you understand Bayes' Theorem well enough, you don't need it.

==Trivia==
*When this comic came out, the title text was only "P((B", and the comic itself linked to [https://xkcd.com/2545/A) A)] or [https://xkcd.com/A) A)] (depending on where the comic was viewed from) and the "Black Lives Matter" image in the header was replaced by "(A", but this was quickly corrected. 
**See this [https://web.archive.org/web/20211122212442/https://xkcd.com/2545/ archived] version.
*It turns out that it is the notation that messes with the home page as it also messes with this wiki. 
**In this [https://www.explainxkcd.com/wiki/index.php?title=2545:_Bayes%27_Theorem&oldid=221182 version] of this page, the [https://www.explainxkcd.com/wiki/index.php?title=2545%3A_Bayes%27_Theorem&type=revision&diff=221182&oldid=221181 correct title text] has been entered, but it still looked the same so everything from behind the first "|" fails to show.
**Now this has [https://www.explainxkcd.com/wiki/index.php?title=2545%3A_Bayes%27_Theorem&type=revision&diff=221183&oldid=221182 been fixed] using the <nowiki><nowiki></nowiki> format.
***Seems like [[Randall]] made an exploit on himself like [[Mrs. Roberts]] did in [[327: Exploits of a Mom]].
***This is extra funny since [[Blondie]], is both sometimes used for Mrs Roberts and for Miss Lenhart from this comic.

{{comic discussion}}

[[Category:Comics featuring Miss Lenhart]]
[[Category:Statistics]]