Wikipedia:Articles for deletion/Normally distributed and uncorrelated does not imply independent

The following discussion is an archived debate of the proposed deletion of the article below. Please do not modify it. Subsequent comments should be made on the appropriate discussion page (such as the article's talk page or in a deletion review). No further edits should be made to this page.

The result was no consensus defaulting to Keep and w/o prejudice to a future renomination. Normally I would relist at least once but in this case we have a discussion with significant participation and opinions that are all over the place. I do not believe a relist would end with consensus. Ad Orientem (talk) 03:00, 10 July 2018 (UTC)[reply]

Normally distributed and uncorrelated does not imply independent

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Normally distributed and uncorrelated does not imply independent (edit | talk | history | protect | delete | links | watch | logs | views) – (View log · Stats)
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Per WP:OR / WP:NOTESSAY, essentially. The whole thing boils down to "A Pearson's correlation coefficient of 0 does not mean variables are independent" (adding the normal distribution is a bit of a red herring). I do not see sources sufficient to establish that this very topic is anywhere close notable. It might be mentioned in a lot of places (e.g. ref 2) as a common student mistake, but not as an encyclopedic subject worthy of careful study.

If not kept, some cleanup is needed, as there are quite a few incoming links. A selective merge to the PCC article or the PCC section of correlation and dependence or might be workable. TigraanClick here to contact me 09:33, 2 July 2018 (UTC)[reply]

You seem to be missing the same point that the nominator missed: If this were to be merged into another article, it should be the articles on the normal distribution and the multivariate normal distribution. Nothing about the normal distribution need be included if they only point were to show that uncorrelatedness does not entail independence, but this article is explaining a fact about the normal distribution, not a fact about the relationship between independence and uncorrelatedness. Michael Hardy (talk) 00:36, 3 July 2018 (UTC)[reply]
Somewhere in Multivariate_normal_distribution#Joint_normality, then? TigraanClick here to contact me 07:33, 3 July 2018 (UTC)[reply]
Pages that have been merged to other articles should almost never be deleted (WP:MAD) Qwfp (talk) 18:55, 3 July 2018 (UTC)[reply]
  • Oppose deletion. This is an insightful article that highlights the difference between joint normality and marginal normality. And no, bringing in normality is not a red herring: the whole point is that while joint normality plus no correlation does imply independence (something that does not hold for non-normal distributions), marginal normality plus no correlation does not imply independence. The examples serve to illustrate how this can be. Loraof (talk) 15:00, 2 July 2018 (UTC)[reply]
    Your point (repeated below by others) duly noted. However, that makes me doubt even more of the subject's notability; "PCC=0 is not independence" is certainly more interesting in the grand scheme of things[original research?] than (joint normality + uncorrelated) <-> (joint normality + independence) (of course, my subjective judgement on that point is not worth zilch, what matters are the refs). We are not a repository of interesting mathematical facts. TigraanClick here to contact me 07:33, 3 July 2018 (UTC)[reply]
    @Loraof: Several people have advocated merging instead; do you oppose that as well? If so, it might be clearer to just change to a keep, but if not, it might be worth saying so. –Deacon Vorbis (carbon • videos) 14:49, 4 July 2018 (UTC)[reply]
@Deacon Vorbis: If it is merged, it should be to the currently brief section Multivariate normal distribution#Two normally distributed random variables need not be jointly bivariate normal. I’m not opposed in principle to doing this, but I’m concerned that the multivariate normal distribution article is already very long. Loraof (talk) 19:18, 4 July 2018 (UTC)[reply]
Note: This discussion has been included in the list of Mathematics-related deletion discussions. MarginalCost (talk) 16:38, 2 July 2018 (UTC)[reply]
  • Merge with another relevant article. The article is very nicely written and very helpful. I strongly oppose deletion. Goharshady (talk) 17:01, 2 July 2018 (UTC)[reply]
  • Keep. An important family of counterexamples, properly sourced. There are multiple textbook sources on exactly this topic [1] [2] [3] [4] [5]. —David Eppstein (talk) 17:21, 2 July 2018 (UTC)[reply]
    I am not disputing this is well-sourced (as in WP:V), I am disputing the notability (as a standalone article) of that bit of math. I do not think multiple short mentions in specialized textbooks demonstrate notability, per the "in-depth" requirement of WP:GNG. You might well say that nothing else could be expected for a run-of-the-mill theorem and wishing for more is an impossible bar to pass, but is that not an indication of non-notability? TigraanClick here to contact me 07:33, 3 July 2018 (UTC)[reply]
  • Keep. Seems like a reasonable short encyclopedia article on some interesting mathematical objects. It would probably also be okay as a section of some other article, but it's hard to see a pressing need to merge. --2601:142:3:F83A:95A9:DA87:64C2:9A47 (talk) 18:01, 2 July 2018 (UTC)[reply]
  • Keep. Tigraan's statement about a "red herring" shows that Tigraan has no clue what this article is about. In a multivariate normal distribution, linear combinations of the components are independent if they are uncorrelated; this article shows that for each component to be normally distributed is not enough; rather, joint normality is needed. People should wait until they understand what an article says before nominating it for deletion. Michael Hardy (talk) 00:34, 3 July 2018 (UTC)[reply]
    See above. Whether I have no clue what this article is about is hardly a rebuttal about my concerns of notability. TigraanClick here to contact me 07:33, 3 July 2018 (UTC)[reply]
  • Comment. This article makes a point about the nature of the normal distribution, not about the relationship between uncorrelatedness and independence. I wonder if it should be dumbed down somewhat to make that clearer. I would never have guessed that it could be misunderstood in the particular way in which the nominator and one of the other posters above have misunderstood it. Michael Hardy (talk) 00:57, 3 July 2018 (UTC)[reply]
  • Delete - This is not an encylopaedia topic, it's an essay or perhaps something for wikibooks. Anything relevant can be put into say Pearson coefficient. The merits of knowing this does not in of itself make it a notable or appropriate topic for an encyclopaedia. -mattbuck (Talk) 10:09, 3 July 2018 (UTC)[reply]
This material certainly does not belong at Pearson coefficient, and the suggestion that it could makes me more sympathetic to Michael Harry's comment than I was before. --2601:142:3:F83A:8985:D0DD:B024:F94C (talk) 11:54, 3 July 2018 (UTC)[reply]
Then some other appropriate article about a theorem, method or technique (thanks JohnBlackburne). Just because you can come up with an example of something does not make it worthy of its own article. -mattbuck (Talk) 19:04, 3 July 2018 (UTC)[reply]
Even this more general wording is wrong: if it were to be merged somewhere, it would be to an article about the (multivariate) normal distribution. Are you sure you have the competence necessary to judge whether this is of encyclopedic importance? Lots of "examples of something" are. --2601:142:3:F83A:181:62BC:A65:DA3F (talk) 21:05, 3 July 2018 (UTC)[reply]
@Mattbuck: You have missed the article's point entirely. The examples in this article are NOT notable among examples that are of importance in the topic of correlation, but they ARE relevant to the topic of the normal distribution, because JOINT normality plus uncorrelatedness does entail independence. Michael Hardy (talk) 21:12, 4 July 2018 (UTC)[reply]
  • delete. Not a named theorem, method or technique, not independently notable by any stretch. Just examples of Correlation and dependence, and not especially interesting ones being very contrived. There’s nothing special about the normal distribution in this, no evidence in the article that there is.--JohnBlackburnewordsdeeds 12:00, 3 July 2018 (UTC)[reply]
Once again, I feel like this validates Michael Hardy's comments. You clearly have failed to understand what the article is about. --2601:142:3:F83A:611C:BD4F:C063:4BF2 (talk) 12:42, 3 July 2018 (UTC)[reply]
  • John, please take a look at my comment above. That comment, and the article, point out that the joint normal distribution has a property that is not generally true across distributions, but the marginal normal distributions do not have this property. It’s entirely about the joint vs. marginal normal distributions. Perhaps the second sentence in the article (which I have now deleted) led some readers astray by going off on a brief irrelevant tangent about another distribution. Loraof (talk) 14:48, 3 July 2018 (UTC)[reply]
@JohnBlackburne: You have missed the point of the article if you think there's nothing special about the normal distribution here. The topic of the article is NOT notable as a comment about correlation and dependence, but it IS notable as a fact about the normal distribution, because JOINT normality plus uncorrelatedness DOES entail independence. Michael Hardy (talk) 21:08, 4 July 2018 (UTC)[reply]
  • strong keep. Counterexamples are a perfectly reasonable topic for a wikipedia page and this is a counterexample to the supposed theorem 'Normally distributed and uncorrelated implies independent'. Having said that, it might be nice to make such counterexamples easier to find; Game without a value, for example, is a counterexample to the supposed theorem 'every zero-sum game has a value', but it's not clear to me how a user might discover this by searching. The page is well-written and informative about notoriously difficult statistical concept independence. Robinh (talk) 20:50, 3 July 2018 (UTC)[reply]
What would be nice is a short subsection in multivariate normal distribution with a {{main}} link to this article. Indeed, that's what we have now. --2601:142:3:F83A:181:62BC:A65:DA3F (talk) 21:07, 3 July 2018 (UTC)[reply]
  • Keep I've been meaning to read it since it's something I want to understand better. That already shows it's useful. I don't agree with the notion that common student mistakes aren't worth writing up in articles. 0.999... is a featured article with a lot of traffic and a long history, giving a dozen or so proofs that 0.999.... is equal to 1, because so many people are confused over this issue. Simpson's paradox is another of these probability counterexamples that's similarly of good value. And Counterexamples in Topology is a famous book that spawned a lot of similar books in other topics like analysis. Added: also, based on Multivariate_normal_distribution#Two_normally_distributed_random_variables_need_not_be_jointly_bivariate_normal mentioned by 2601:142:3:F83A:181:62BC:A65:DA3F, splitting it out looks ok per WP:Summary style. 173.228.123.166 (talk) 07:06, 4 July 2018 (UTC)[reply]
    But 0.999... has an incomparable level of referencing. For instance I would assume (I cannot access it) that Beswick, Kim (2004). "Why Does 0.999... = 1?: A Perennial Question and Number Sense". Australian Mathematics Teacher. 60 (4): 7–9. discusses it in an encyclopedically-compatible way (i.e. shows notability). Same for Simpson's paradox - this article is entirely devoted to a meta-analysis of the paradox (not just a proof of the math).
I do not think anyone is arguing that examples and counterexamples should be excluded on principle from math articles - the question here is whether we need a standalone article. (And WP:ITSUSEFUL does not help.) We should not use summary style when the spin-off is not notable is itself (WP:AVOIDSPLIT). TigraanClick here to contact me 08:11, 4 July 2018 (UTC)[reply]
It's not a big deal to me whether it's a separate article but what you nominated the article for was deletion. If we merge it to another article while not losing any significant content then ok, but as someone else said, I don't see much urgency to it. Note also that the purpose of the those policies you keep mentioning is to improve the encyclopedia. So I'm unimpressed by arguments from wiki policy for any particular action, unless they can be supported by an explanation of how that action improves the encyclopedia directly. I haven't seen any attempt from you to do that so far. Following policy for its own sake is basically the definition of bureaucracy, and arguing purely from policy is wikilawyering, that should almost always be seen unfavorably. Policy is not an axiom system whose consequences are theorems. It's more like a low order regression-fit of past experience whose suggestions are at best approximate in any situation, and at worst completely off. So it always has to be checked against specific cases before applying it, if there is any doubt at all.

I therefore find your line of argument distasteful rather than persuasive. The most mathematically knowledgeable contributors here all seem to want to keep the article or at least preserve its contents, and that's good enough for me. 173.228.123.166 (talk) 15:20, 4 July 2018 (UTC)[reply]

what you nominated the article for was deletion: Yes and no. If you know a better place than AfD to discuss cases where it is unclear whether the endgame is deletion, merge, redirect and to which target, please let me know, because I am not aware of one; WP:RM is pretty much a binary "move or not move", and WP:RFD rarely addresses historical content of the redirects.
I'm unimpressed by arguments from wiki policy for any particular action, unless they can be supported by an explanation of how that action improves the encyclopedia directly. You are reversing the burden of proof of WP:IAR / "guidelines are not absolute" here. If you want to argue IAR (i.e. that policy says to do X but the best outcome for the encyclopedia is Y), the onus is on you to demonstrate that special circumstances apply, or (reusing your metaphor) that the current datapoint does not fall on the fit line. Policies exist for a reason, and we do not rediscuss them at every application.
The most mathematically knowledgeable contributors here all seem to want to keep the article or at least preserve its contents, and that's good enough for me. - Well, that's an argument from authority, but more to the point mathematical competence is weakly correlated to Wikipedia article content handling. That line of reasoning leads straight to "we should defer to homeopaths/crystal healers/dowsers when it comes to content about homeopathy/crystal healing/dowsing", which is not going to happen and fortunately so. A better argument to make would have been "long-time Wikipedia editors with an interest in the subject topic want to keep" - in which case the argument of authority follows from Wikipedia tenure, not mathematics directly - but it still is fairly weak. TigraanClick here to contact me 11:27, 5 July 2018 (UTC)[reply]
This isn't the place for meta-philosophy but I think you are a little confused. IAR (simplified) is when you have an edit that's against policy but you decide to make it anyway because it's a good edit that improves the encyclopedia. The complementary situation, where your edit is allowed by policy but you decide not to make it because it's a lousy edit that doesn't improve the encyclopedia, is not IAR but is just common sense. If an edit doesn't improve the encyclopedia you shouldn't make it. It's never against policy to not make an edit. Therefore the only sound way to justify a proposed edit when people are unconvinced is to explain how it improves the encyclopedia, not what policy says about it. And I'd call it bloody obvious (WP:CIR) that in a question of math exposition (which is what this is), the views of the knowledgeable math editors have to carry greater weight than those of editors who are merely interested in the subject but don't understand it. "Wikipedia content handling" is supposed to serve the goal of exposition, not the other way around.

The usual place to propose an article merge is on the article talk page, not AfD. You can use the {{merge from}} talkpage template for the purpose. The talk page of the relevant wikiproject (WT:WPMATH for this) is probably also a good place to leave a notice. 173.228.123.166 (talk) 22:36, 5 July 2018 (UTC)[reply]

Note: This discussion has been included in the list of Science-related deletion discussions. Tyw7  (🗣️ Talk to me • ✍️ Contributions) 10:31, 4 July 2018 (UTC)[reply]
Note: This discussion has been included in the list of Mathematics-related deletion discussions. Tyw7  (🗣️ Talk to me • ✍️ Contributions) 10:31, 4 July 2018 (UTC)[reply]
  • Merge to Multivariate normal distribution. We don't generally have articles on false implications that admit unnamed counterexamples. If an article has to be named "A and B doesn't imply C", then it's a pretty good bet that the information within should be kept at some other appropriate place. For example, we have an article for Tychonoff plank and we can mention that it serves as a counterexample to various implications, but we don't keep articles for any of those particular implications. –Deacon Vorbis (carbon • videos) 14:20, 4 July 2018 (UTC)[reply]
  • keep Is sourced and is long enough that merging doesn't make sense. OR issues are not persuasive due to sourcing. JoshuaZ (talk) 17:45, 4 July 2018 (UTC)[reply]
  • Comment. Quite a few years ago there used to be math-related deletion discussions in which lots of people who had no competence whatsoever in the field would pontificate and talk down to mathematicians about the subject. One of those was a 19-year-old who said he had NEVER heard of a theorem, proof, formula, etc. being called "elegant" (that word is an over-used cliche in reference to mathematics, in my view) and therefore it doesn't make sense to use that word, and furthermore he said no Wikipedia article should be about a topic in mathematics, and he told me I needed his assistance if I could not even understand those points. I thought that might be a thing of the past. But here the nominator himself and at least two others posters have entirely missed two simple points that make me think maybe we should dumb down the article to make those points clearer to naive readers:
(1) The examples in this article are not particularly interesting as examples to illustrate the relationship between correlation and dependence. There are better examples for that purpose. They are better because they are simpler.
(2) However, these examples are of interest to understand something about the normal distribution: Despite the fact that JOINTLY normally distributed random variables are indeed independent if they are uncorrelated, nonetheless that conclusion is NOT true of MARGINALLY normally distributed random variables. Illustrating that point is what the examples are for.
If there is some article into which this should be merged, it would be about the multivariate normal distribution, not about correlation. Michael Hardy (talk) 21:23, 4 July 2018 (UTC)[reply]
@Michael Hardy: Having read your previous comments, I agree about the better merge target.
In the interest of keeping the debate civil and on-point, could you please say whether, in your opinion, the topic described in the current article is notable (as in standalone-article-worthy), based either on current sourcing or other sources to specify? TigraanClick here to contact me 11:38, 5 July 2018 (UTC)[reply]
I voted above to keep the article. I think it could also be merged into Multivariate normal distribution. Michael Hardy (talk) 20:28, 5 July 2018 (UTC)[reply]
You did, but I do not see you having addressed the notability issue. TigraanClick here to contact me 09:21, 6 July 2018 (UTC)[reply]
  • Merge, with trimming. The material is valuable but the title is not the sort of thing we generally have whole WP articles about. WP:NOUN is not just a stylistic point; it's the sort of thing that makes good encyclopedia articles. If you try to make the title into a noun phrase, say Non-entailment of independence from zero correlation of normal variates, it just doesn't work at all. This is usually a decent indication — not as a rigid rule, but at least as a rule of thumb — that the topic doesn't make a good article. --Trovatore (talk) 23:25, 5 July 2018 (UTC)[reply]
    • How about “Correlation and independence in marginal normal distributions”? Loraof (talk) 16:06, 6 July 2018 (UTC)[reply]
      • Hmm, that is a bit better title, I agree. I guess on the merits I also think it's too small a point for a standalone article, independent of the title.
        It's not surprising by itself that zero correlation should fail to imply independence; that's a more general point. Michael's point is that zero correlation does imply independence for joint normal distributions, so it's possible that you could get confused about whether it's true just given that the individual random variables are separately normally distributed.
        But the result for joint normal distributions is just a simple consequence of the fact that those distributions are completely determined by their mean and covariance, and it so happens that a covariance matrix with zero correlation also determines a joint-normal distribution in which the individual variates are independent.
        So it seems to me that there is nothing especially surprising or worthy of calling out as a separate article here, just a refutation of a mistake that statistics students might be prone to make. We don't usually write standalone articles just for that sort of thing, unless it's a mistake that has raised a special ruckus for some reason (like 0.999...). --Trovatore (talk) 20:02, 6 July 2018 (UTC)[reply]
  • Keep well-sourced \\\Septrillion:- ~~‭~~10Eleventeen 08:02, 6 July 2018 (UTC)[reply]
  • Keep or Merge to the most appropriate article on the meaning(s) of statistical independence. Clearly the notions of joint versus marginal normal distributions remain as difficult to grasp as they were when I was teaching them forty-some years ago …! While the article's title is too clumsy to ever grace a best-seller, if that's the best way to describe the topic, so be it. But its very awkwardness suggests that there's something more fundamental involved; and that something, I believe, is "statistical independence". yoyo (talk) 17:14, 6 July 2018 (UTC)[reply]
    • Nothing more fundamental is involved, since as I suggested above, the title could be changed to the unawkward “Correlation and independence in marginal normal distributions”. Loraof (talk) 17:57, 7 July 2018 (UTC)[reply]
  • Merge to Correlation and dependence, as suggested above. This is a corollary to an existing topic, not a topic in itself. Definitely useful material, but does not make a standalone article. --Elmidae (talk · contribs) 18:10, 6 July 2018 (UTC)[reply]
This is a completely inappropriate merge target, as has been explained several times above! --128.164.177.55 (talk) 20:31, 6 July 2018 (UTC)[reply]
Seems fine to me, but I'm not bothered if the target is a different article. There's plenty of candidates. --Elmidae (talk · contribs) 10:22, 7 July 2018 (UTC)[reply]
I do not understand how people who fail to understand what the article is about, and to follow a discussion about it, believe that they can have a sensible opinion about whether its subject is notable or not! --2601:142:3:F83A:716E:8F86:6A20:1BE3 (talk) 13:06, 7 July 2018 (UTC)[reply]
Being a modeler, I understand it just fine, thanks. In my judgement it fits well with that topic. Unlike you I am however not going to blow a capillary if it is integrated into any one of a number of other primary articles on statistical independence, correlation, or the normal distribution. --Elmidae (talk · contribs) 13:18, 7 July 2018 (UTC)[reply]
You are not "blowing a capilary" (by the way, let me leave this here) because you are not in the situation of having to explain to someone that a short article on the eating habits of ''Pantera leo'' should not be merged into an article on Cuisines of Central Africa. If we did the merge you suggest, any reasonable editor of the target article would immediately remove it as off-topic. Do you understand this? If your substantive grasp here is that weak, on what basis should anyone value your !vote? --2601:142:3:F83A:E1C6:E1B2:1AC1:FC7E (talk) 13:30, 7 July 2018 (UTC)[reply]
Huh, italics don't work inside Wikilinks -- has that always been true? --2601:142:3:F83A:E1C6:E1B2:1AC1:FC7E (talk) 13:31, 7 July 2018 (UTC)[reply]
As I said, there are other valid targets, as long it gets merged. You are welcome to continue raging about that particular choice of merge target; not going to respond to the histrionics any further. --Elmidae (talk · contribs) 15:45, 7 July 2018 (UTC)[reply]
I would ask the closing admin to take into account this user's obvious lack of WP:COMPETENCE when evaluating the consensus here. -2601:142:3:F83A:38D7::BE37:3E0B:C907 (talk) 00:35, 8 July 2018 (UTC)[reply]
I would ask the hyperventilating editor to stop confusing "differing assessment" with "WRONG!!!". Amazing how everyone who disagrees with you lacks competence... it must be very lonely at the top... --Elmidae (talk · contribs) 07:58, 8 July 2018 (UTC)[reply]
Yes, well, if being a complete asshole while refusing to address the substantive issue makes you feel better, there is nothing that I can do to stop you. Nevertheless, the opinions of a person who behaves like that should be given 0 weight in any discussion of technical issues. --2601:142:3:F83A:7CB5:5BF:7962:D897 (talk) 23:24, 8 July 2018 (UTC)[reply]
  • Comment: The examples in the article seem to me to have been contrived specifically for the purpose of demonstrating that dependence without correlation is possible. This leaves me doubtful that the possibility of dependence without correlation has any "real world" relevance, other than when the author of a study is suspected of cherry-picking the definitions of variables. NeonMerlin 20:40, 9 July 2018 (UTC)[reply]
    • I doubt it makes very much difference to the current discussion whether or not the fact has real-world significance. It's a refutation of a possible error on the part of students learning the material. My view is that this point does deserve to be treated, but probably in less detail and probably not in its own article. --Trovatore (talk) 22:00, 9 July 2018 (UTC)[reply]
(edit conflict) The examples are constructed to make it easy to verify that they have the relevant properties. (Of course.) It is not clear what "real world" you speak of, but in the real mathematical world they have exactly the following relevance: their existence indicates that a certain implication (two random variables are known to be marginally normal and uncorrelated; therefore they are independent) may *not* be employed without verifying additional hypotheses (that the variables are jointly normal). This kind of example (contrived for easy verification) is extremely common in mathematics textbooks. --2601:142:3:F83A:7D60:3341:364B:EE37 (talk) 22:03, 9 July 2018 (UTC)[reply]
The above discussion is preserved as an archive of the debate. Please do not modify it. Subsequent comments should be made on the appropriate discussion page (such as the article's talk page or in a deletion review). No further edits should be made to this page.
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