Talk:Bayes' theorem

Interpretations Image

The assassin version of the image is literally an inferior visualization, and less intuitive. Specifically, it is less visually apparent that a character is “sus” or “an assassin” compared to having a beard, or wearing glasses. Quite frustrating that people are happy to argue for something that harms the transmissible of knowledge because it’s funny. Uwuo (talk) 04:10, 22 June 2023 (UTC)

@Uwuo: Though mental characteristics are harder to illustrate than physical ones, suspicion/guilt is a far better application of Bayes' theorem than beard/glasses, which has no causal relationship. Cheers, cmɢʟee⎆τaʟκ 00:15, 2 July 2023 (UTC)

@Cmglee: Why do we need any visible attributes at all? Can't we just stick to plain numbers? What additional information do six face icons contain that a plain digit 6 doesn't? All the examples in the #Examples section work perfectly with digits, why can't #Interpretations do the same? --CiaPan (talk) 18:40, 2 July 2023 (UTC)

I support the use of a visual representation. I support the present visual over the beards and glasses visual because beards and glasses have no connection whereas being suspicious and being an assassin are believably connected. However, I also support improving the current diagram and even replacing it with something else, so long as the two variables have some connection. In the current diagram, being an assassin is indicated by a dagger. That one makes sense. Being suspicious is indicated by what? Is it a bushy eyebrow? Maybe an eyepatch would be better. Constant314 (talk) 18:49, 2 July 2023 (UTC)

Thanks, @Constant314. https://onlinelibrary.wiley.com/doi/abs/10.1111/jopy.12396 concludes that "distinctive eyebrows reveal narcissists' personality to others, providing a basic understanding of the mechanism through which people can identify narcissistic personality traits with potential application to daily life."

I've considered changing it to having some bloodstains (also starts with "B") I'll change it if you concur. Cheers, cmɢʟee⎆τaʟκ 19:15, 2 July 2023 (UTC)

I like that. See a man with blood stains on him and it is more likely he is an assassin. Constant314 (talk) 21:30, 2 July 2023 (UTC)

What have i stumbled upon LuckTheWolf (talk) 08:57, 9 October 2023 (UTC)

It does seem needlessly distracting (if the reader gets the Among Us reference) or confusing (if they don't) to have the assassins also wearing little visors and backpacks.

Maybe the suspicion marker could be a cloak, to go with the dagger? Belbury (talk) 17:13, 24 April 2024 (UTC)

While it may make no difference to you, some people are more visually minded and find it easier to understand or learn concepts when given concrete representations. Cheers, cmɢʟee⎆τaʟκ 19:10, 2 July 2023 (UTC)

@Uwuo this is why we can't have nice things Mihnea Liliac (talk) 17:04, 22 October 2025 (UTC)

Marginal probability vs. evidence

This may have been asked before. The article says, P(B) is called marginal probability. But in the top right hand overview box, the very first line reads Posterior = Likelihood × Prior ÷ Evidence. So here, P(B) is called evidence. I am aware that the word evidence is used differently in the article. Nevertheless there is an inconsistency.

D. S. Sivia in his book Data Analysîs writes on page 6 (his notation is (p(data|I) instead of p(B)): In some situations, like model selection, this term plays a crucial role. For that reason, it is sometimes given the special name of evidence. This crisp single word captures the significance of the entity, as opposed to older names, such as prior predictive and marginal likelyhood ... Such a central quantity ought to have a simple name, and evidence has been assigned no other technical meaning (apart from as a colloquial synonym for data).

The last parenthesis is just the problem :-) Still, I find Sivia's argument for the name evidence very good. Herbmuell (talk) 10:18, 11 November 2025 (UTC)

Genetics

Recently a section on genetics was largely deleted. I agree that it is too long for an article such as this, but looking through the literature it appears to me that Bayesian Analysis in Genetics is covered independently enough as a topic to warrant an article, so I have moved the deleted text here should someone in the future wish to use it in such an article.ເສລີພາບ (talk) 23:54, 1 March 2026 (UTC)

Extended content

In genetics, Bayes' rule can be used to estimate the probability that someone has a specific genotype. Many people seek to assess their chances of being affected by a genetic disease or their likelihood of being a carrier for a recessive gene of interest. A Bayesian analysis can be done based on family history or genetic testing to predict whether someone will develop a disease or pass one on to their children. Genetic testing and prediction is common among couples who plan to have children but are concerned that they may both be carriers for a disease, especially in communities with low genetic variance.

Hypothesis	Hypothesis 1: Patient is a carrier	Hypothesis 2: Patient is not a carrier
Prior Probability	1/2	1/2
Conditional Probability that all four offspring will be unaffected	(1/2) ⋅ (1/2) ⋅ (1/2) ⋅ (1/2) = 1/16	About 1
Joint Probability	(1/2) ⋅ (1/16) = 1/32	(1/2) ⋅ 1 = 1/2
Posterior Probability	(1/32) / (1/32 + 1/2) = 1/17	(1/2) / (1/32 + 1/2) = 16/17

Above is an example of a Bayesian analysis table for a female's risk for an X-linked recessive disease based on the knowledge that the disease is present in her siblings but not in her parents or any of her four children. Based solely on the status of the subject's siblings and parents, she is equally likely to be a carrier as to be a non-carrier (this likelihood is denoted by the prior hypothesis). The probability that the subject's four sons would all be unaffected is 1/16 (1⁄2⋅1⁄2⋅1⁄2⋅1⁄2) if she is a carrier and about 1 if she is a non-carrier (this is the conditional probability). The joint probability reconciles these two predictions by multiplying them together. The last line (the posterior probability) is calculated by dividing the joint probability for each hypothesis by the sum of both joint probabilities.

Parental genetic testing can detect around 90% of known disease alleles in parents that can lead to carrier or affected status in their children. Cystic fibrosis is a heritable disease caused by an autosomal recessive mutation on the CFTR gene, located on chromosome 7.

Here is a Bayesian analysis of a female patient with a family history of cystic fibrosis (CF) who has tested negative for CF, demonstrating how the method was used to determine her risk of having a child born with CF: because the patient is unaffected, she is either homozygous for the wild-type allele, or heterozygous. To establish prior probabilities, a Punnett square is used, based on the knowledge that neither parent was affected by the disease but both could have been carriers:

Mother Father	W Homozygous for the wild- type allele (a non-carrier)	M Heterozygous (a CF carrier)
W Homozygous for the wild- type allele (a non-carrier)	WW	MW
M Heterozygous (a CF carrier)	MW	MM (affected by cystic fibrosis)

Given that the patient is unaffected, there are only three possibilities. Within these three, there are two scenarios in which the patient carries the mutant allele. Thus the prior probabilities are 2⁄3 and 1⁄3.

Next, the patient undergoes genetic testing and tests negative for cystic fibrosis. This test has a 90% detection rate, so the conditional probabilities of a negative test are 1/10 and 1. Finally, the joint and posterior probabilities are calculated as before.

Hypothesis	Hypothesis 1: Patient is a carrier	Hypothesis 2: Patient is not a carrier
Prior Probability	2/3	1/3
Conditional Probability of a negative test	1/10	1
Joint Probability	1/15	1/3
Posterior Probability	1/6	5/6

After carrying out the same analysis on the patient's male partner (with a negative test result), the chance that their child is affected is the product of the parents' respective posterior probabilities for being carriers times the chance that two carriers will produce an affected offspring (1⁄4).

Bayesian analysis can be done using phenotypic information associated with a genetic condition. When combined with genetic testing, this analysis becomes much more complicated. Cystic fibrosis, for example, can be identified in a fetus with an ultrasound looking for an echogenic bowel, one that appears brighter than normal on a scan. This is not a foolproof test, as an echogenic bowel can be present in a perfectly healthy fetus. Parental genetic testing is very influential in this case, where a phenotypic facet can be overly influential in probability calculation. In the case of a fetus with an echogenic bowel, with a mother who has been tested and is known to be a CF carrier, the posterior probability that the fetus has the disease is very high (0.64). But once the father has tested negative for CF, the posterior probability drops significantly (to 0.16).

Risk factor calculation is a powerful tool in genetic counseling and reproductive planning but cannot be treated as the only important factor. As above, incomplete testing can yield falsely high probability of carrier status, and testing can be financially inaccessible or unfeasible when a parent is not present.

References

*Rational* degree of belief

I have inserted "rational" to emphasize the non-arbitrary nature of Bayesian "belief". This is per Jaynes (and emphasized by him), our reference #32. However, I do not know how to show that reference (appearing as superscript "[32]") properly in the text. If someone can, please do. Jmacwiki (talk) 19:08, 26 April 2026 (UTC)

Not Monty Hall

The "Recreational mathematics" section says that Bayes Theorem can give us the resolution to the Monty Hall paradox (and others). However, this paradox isn't caused by misunderstanding how to compute. It's caused by misunderstanding what the rules of the game are: They are not the same in nature ("randomly" revealing items) as they are in the game (Monty Hall will never reveal the car: a non-random choice).

Bayes doesn't tell how to recognize what rules are in play, just what to compute according to those rules.

I suggest deleting the wording about this problem. However, I am not familiar with the other examples. Do any of them have the same problem? Jmacwiki (talk) 23:23, 26 April 2026 (UTC)

That the many wrong opinions about the Monte Hall problem are caused by misunderstanding the rules is a very strong and, I think, unsupportable claim. Bayes Theorem does give the correct answer to the problem: a problem that even many people with PhDs have got wrong. So the section definitely belongs in this article. Your claim about Monte Hall problems happening "in nature" (with slightly different rules to the TV show) is bizarre and I have no idea what situation you are referring to. A game show in which contestants can win a car by choosing one of three doors is an artificial, not natural, phenomenon. MartinPoulter (talk) 12:33, 27 April 2026 (UTC)

Misunderstanding the rules does contribute. For example does Monty always open a door and offer a swap. When he does open one of the other doors, is it a random choice or is it always the item with the lessor value? But I also agree that people get it wrong even when the rules are explicit. Constant314 (talk) 13:03, 27 April 2026 (UTC)

The given source (Rosenhouse's book) talks about people misunderstanding the rules and poor statements of the problem that don't make clear how Monty chooses a door, and also how the problem confuses people even when correctly stated. Stepwise Continuous Dysfunction (talk) 19:58, 27 April 2026 (UTC)

All the article says is Bayes' rule and computing conditional probabilities provide a method to solve a number of popular puzzles, such as the Three Prisoners problem, the Monty Hall problem, ...

Obviously you cannot apply Bayes until you settle the rules. Basically, the Monty Hall problem is a family of puzzles, each with its own rules. Plenty to squabble about there, but once you select a rule set, you use Bayes. I don't see any need to change the article. Constant314 (talk) 21:26, 27 April 2026 (UTC)

To be clear, I do not see a need either. Stepwise Continuous Dysfunction (talk) 00:23, 28 April 2026 (UTC)

Our current text says, "Bayes' rule and computing conditional probabilities provide a method to solve a number of popular puzzles, such as ... the Monty Hall problem...." This is misleading to the reader.

The reason those are "popular puzzles" is, in my experience (as both a frequent user of Bayes and a teacher of prob & stats), precisely because of the paradox that the game show's rules are not the ones many (most?) people believe are in use. The latter rules are what I labeled "in nature": Select 1 of 3 rocks covering a treasure, wait for nature to remove one of the rocks with equal probability -- the essence of the paradox -- and compute what the expected return is.

Bayes helps equally in either set of rules. But it doesn't solve the puzzle because it doesn't do anything to answer the paradox: It doesn't help anyone determine that Monty Hall chooses non-randomly. It's a "popular puzzle" because it's counterintuitive, i.e., because of the paradox.

We might equally say "Arithmetic provides a method...", which would be exactly as true and exactly as misleading. If we don't delete outright, can we at least find wording that doesn't mislead the reader? Jmacwiki (talk) 21:48, 29 April 2026 (UTC)

Where in the great mass of text, going back decades, that talks about the Monty Hall problem, is this "natural" variant that involves erosion of rocks? Your experience isn't a citable source, especially if it contradicts what's demonstrated in published sources: that even intelligent, educated people are convinced of the wrong answer even when understanding the correct rules of the problem. MartinPoulter (talk) 10:45, 30 April 2026 (UTC)

That whole recreational math section adds almost nothing to the article. It could be removed with almost no loss. It is just a factoid with no explanations. Constant314 (talk) 16:24, 30 April 2026 (UTC)

One person's "factoid" is another's "summary style"... I don't think it adds a lot to the page, and I wouldn't weep over its omission, but I do think it's a net positive. Stepwise Continuous Dysfunction (talk) 19:55, 30 April 2026 (UTC)

Arguably, the section could support a point about psychology -- but on some psych page, not a Bayes Theorem page. Jmacwiki (talk) 21:53, 1 May 2026 (UTC)

People find it paradoxical because nature (and plenty of non-natural random processes) do not behave as Monty Hall does; their intuition leads them to infer a different set of rules than he plays.

Whether there is a published study is a red herring, since we would not be making a statement in the article -- we are discussing whether to say anything at all. Jmacwiki (talk) 21:50, 1 May 2026 (UTC)

This short paragraph is concise and referenced. The 1989 Cognition article explicitly includes the Three prisoners problem (equivalent to the Monty Hall problem) as a "Bayesian problem of probability". WeyerStudentOfAgrippa (talk) 22:54, 1 May 2026 (UTC)

Agreed, that makes it citable here. I believe it is unhelpful, not uncitable. Jmacwiki (talk) 04:10, 4 May 2026 (UTC)

The source given in the article, and many more out there in the literature, use Bayes' theorem as a tool for stepping through the Monty Hall problem and finding the right answer. Whether or not that is the most persuasive way of getting people to "see the light" and accept the true answer is a different question, and one that is beyond the scope of this article. What matters here is that the Monty Hall problem is one of the many things that Bayes' theorem has verifiably been applied to. Stepwise Continuous Dysfunction (talk) 16:54, 4 May 2026 (UTC)

FWIW, I find this last argument at least somewhat persuasive and reassuring. Thanks. Jmacwiki (talk) 05:03, 8 May 2026 (UTC)