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This question

What does a $\sigma$-field have to do with foundations of probability theory?

was closed as too broad. In the strict sense that's true: there is no precisely stated problem to be solved. But as a query from someone who's not studying mathematics and needs some guidance I thought it reasonable and essayed an answer. That answer was accepted and generally well received, so the deletion surprised me.

(There's also a deleted clever non-answer. I think it should stay.)

I'm posting this here to raise a policy discussion - I think we should welcome this kind of question from this kind of visitor. I'm not concerned about reopenig this particular instance.

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    $\begingroup$ Perhaps, you should vote to reopen, then? $\endgroup$ – Mark McClure Dec 16 '18 at 16:05
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    $\begingroup$ @Ethan Do you think a few paragraphs can answer "What are the foundations of probability theory?" If you believe you answered the two questions that were asked in that post, then I can fly by flapping my arms alone. $\endgroup$ – Namaste Dec 16 '18 at 17:58
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    $\begingroup$ @amWhy I think my answer addresses what the OP needs to know (to start) about "the foundations of probability". A more precise answer to the second question would call for a discussion of countable additivity. I don't think it would help the OP get started reading the text. You don't have to try to fly even if I have convinced you that my answer is appropriate. $\endgroup$ – Ethan Bolker Dec 16 '18 at 18:11
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    $\begingroup$ Your last sentence doesn't make sense. Anyway, there are books written on the foundations of probability theory. If you have convinced yourself that your answer satisfactorily answers the question that takes most a book to do, then I think you need to re-evaluate your... errrr self-assessment?? $\endgroup$ – Namaste Dec 16 '18 at 18:15
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    $\begingroup$ @amWhy I hope you don't think I think my answer developed the foundations of probability. I did try to address what I think the OP needs to know to get past that phrase and start studying NLP. Sorry I mangled the last sentence about whether you could fly. $\endgroup$ – Ethan Bolker Dec 16 '18 at 18:20
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    $\begingroup$ "I hope you don't think I think my answer developed the foundations of probability. I did try to address what I think the OP needs to know to get past that phrase...". How wonderful that you are endowed with the capacity to read other users' minds, e.g., "what I think the OP needs to know..." $\endgroup$ – Namaste Dec 16 '18 at 18:23
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    $\begingroup$ @amWhy (Last response). As a teacher I am used to (beginning) students asking questions that don't quite ask what they need to know. It's part of my job to try to guess that. What you call "mind reading" I call "classroom experience". Worst case (here and in class) is that I fail, and they have to ask again, or differently. Best case I've helped. $\endgroup$ – Ethan Bolker Dec 16 '18 at 18:28
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    $\begingroup$ In this case, I'd suggest you over-estimated your "classroom experience". "Asking again, with a clearer and more precise question" is not a worst case scenario" for the asker. It is an educational experience in which they learn to hone in on the question they intend to ask (if it is indeed less broad than what was posted). $\endgroup$ – Namaste Dec 16 '18 at 18:32
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    $\begingroup$ (cont'd) Mind you, I know that in many places people try to teach students probability theory, including $\sigma$-algebras, WITHOUT first making the students take a course on measure theory. I attended such a course as a freshman myself! The disconnect between pure and applied math. Anyway, I can see why a student in such a situation would want to ask this question. $\endgroup$ – Jyrki Lahtonen Dec 17 '18 at 6:56
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    $\begingroup$ Hello, I'm the author of the "deleted non answer". While I may have focused on the first part of the question ("Foundations of probability theory"), the only thing I didn't include regarding sigma algebras is that if you have an event $E$, then $E$ not happening should be an event, and with two events $E, F$, either of them happening should also be an event. There. That's why we want sigma algebras - no lengthy Wikipedia entry required. From my point count, it would appear that many readers appreciated my response. It must have been those 50 people with a sense of humor. (ctd.) $\endgroup$ – Matthias Dec 17 '18 at 19:41
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    $\begingroup$ (ctd) The pedantry and and pearl clutching displayed in this discussions do make me reconsider my involvement with this site. Maybe math isn't your thing, maybe politics should be. In any case, it's been real – Tah! $\endgroup$ – Matthias Dec 17 '18 at 19:43
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    $\begingroup$ @Matthias I quite enjoyed your answer, especially its unusual, Biblical style. :) I am surprised and dismayed to see it deleted, as I think it did provide an answer to the question, and had a score of +11 at the time of deletion. For instance, here is a similarly unusual answer that is highly upvoted, despite the fact that it is even more indirect (or perhaps, analogical) than yours. $\endgroup$ – André 3000 Dec 17 '18 at 21:45
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    $\begingroup$ In case you are curious, here are the timeline and review task for your answer. In the end, it seems to have been mod-deleted by moderator Aloizio Macedo, so there it can't currently be reopened by regular users. $\endgroup$ – André 3000 Dec 17 '18 at 21:46
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    $\begingroup$ @RobertSoupe I appreciate your concern and would like to put at ease. Of course I cannot speak for others, but let me (as a convinced christian) say that I liked the “clever non-answer”. I believe that God created man in His own image and that consequently human beings are creative. When it concerns science then where does this come forward? In mathematics!! Fully made by mankind and no essential need of external phenomena. Just thinking on its own is enough already to create e.g. things like $\sigma$-algebras. Reading the non-anwer again I saw that resemblance and enjoyed it. $\endgroup$ – drhab Dec 23 '18 at 7:06
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    $\begingroup$ I thought that @Matthias's answer was very good and I too am dismayed that it was deleted. Math exposition does not always have to be dry and formal. The playful style helps some readers to feel that measure theory is not so intimidating. I hope the answer gets undeleted. (Besides, is anyone hurt by having the answer there? It is if nothing else a thoughtful and sincere attempt by someone with a flair for teaching to help people learn math.) $\endgroup$ – littleO Dec 23 '18 at 14:52
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It seems a reasonable (not overly broad) Question to me. I made an edit to improve it. In the original post the motivation is described, reading a claim about the "foundation" of probability theory. The actual book and author were given in the first Comment.

A sympathetic reading of the Question is that the OP wonders why a $\sigma$-algebra is needed to build a theory of probability. I can imagine this thought crossing the minds of many first-year graduate students when first exposed to measure theory, so it is to be expected that folks who stumble into this topic from an applied field (natural language processing) often share this feeling of surprise.

The "overly broad" close reason is described in terms of "if your question could be answered by an entire book". While a request to provide the entire foundation of probability theory would trigger that concern, here the request is more limited: why do we need the elaborate notion of a $\sigma$-algebra for the purpose?

This is something (IMHO) that can be answered in a reasonably definitive way within the limits of the StackExchange format, so I voted to reopen. I also tried to vote to undelete the answer that uses a humorous parody of Genesis to put the $\sigma$-algebra into this context. It is perhaps limited in its success, but I don't think humor always denotes a lack of seriousness. However that Answer was deleted by a moderator, so my undelete vote was rejected.

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    $\begingroup$ The question asked two things: "What are the foundations of probability theory?" (On its own this is too broad of a question to satisfactorily answer in one question,) and "How does $\sigma$-algebra figure in this. That's hardly digestible in one answer. $\endgroup$ – Namaste Dec 16 '18 at 17:56
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    $\begingroup$ @amWhy: What you say is literally true, but it's what I would call an unsympathetic reading of the Question. You fail to mention it, but you rolled back my edits and those of another user who attempted to improve the Question, perhaps to make your criticism of it being "too broad" appear in a better light. $\endgroup$ – hardmath Dec 16 '18 at 18:04
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    $\begingroup$ And you edited perhaps to make your advocacy of it appear in a better light? I think the question was closed as it is currently stated; if the OP wants to alter the question, give them time to do so before paternalistically changing it to what you think they ought to have asked, or aiding an answerer by altering the question to match their insufficient answer. $\endgroup$ – Namaste Dec 16 '18 at 18:08
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    $\begingroup$ @amWhy, I don't see how including content from the OP's own Comment into the body of the Question is "paternalistic". I assure you it was meant to clarify the Question, and was done before I wrote up my analysis (discussion of overly broad policy) above. $\endgroup$ – hardmath Dec 16 '18 at 18:15
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    $\begingroup$ +1 Although broad, I see a lot of value in the question. It is not the typically silly stuff we reject for being too broad. So in this case, I think rejection would be a 'foolish consistency' depriving us of a fairly interesting question. $\endgroup$ – rschwieb Dec 18 '18 at 14:27
  • $\begingroup$ So, @rschwieb You'd be okay with a series of posts asking: (1) What are the foundations of set theory, and (2) how is the axiom of choice involved; or (1) What are the foundations of group theory and (2) how does some guy named Sylow figure into it?; or any other such double-questions which is beyond the scope of what this site intends for Q&A?? $\endgroup$ – Namaste Dec 18 '18 at 16:42
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    $\begingroup$ @amWhy I see no reason to speculate on what I'd do in your hypothetical situations. After all, my discretion is better than binary. My comment is what it is: an evaluation of the case at hand. In this case, discretion tells me that the question meets, or show promise of meeting, a level of quality that we can value. You seem to be implying that I would allow such questions of any quality, which is exactly the opposite of the point I'm trying to make (that discretion is involved.) $\endgroup$ – rschwieb Dec 18 '18 at 16:53
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    $\begingroup$ @amWhy TL;DR I don't see it as an all-or-nothing proposition, as you seem to be trying to box me into. $\endgroup$ – rschwieb Dec 18 '18 at 16:59
  • $\begingroup$ "I think the question was closed as it is currently stated; if the OP wants to alter the question, give them time to do so before paternalistically changing it to what you think they ought to have asked, or aiding an answerer by altering the question to match their insufficient answer." The edits were adding information that the OP gave in comments @amWhy. However, I question why we should be approving edits to questions that are off topic. They are off topic. Therefore they are ineligible for editing because doing so is a waste of everyone's time. -1. $\endgroup$ – The Great Duck Dec 28 '18 at 7:55
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I think we should welcome this kind of question from this kind of visitor.

I strongly disagree.

tl;dr Wikipedia serves this purpose far better than StackExchange. When it doesn't, that usually means answers need to be tailored very specifically to the asker, making them not that useful for others. $\square$

We should assist this kind of visitor to reformulate their question into a more appropriate question if possible. A more appropriate formulation might be a , or, more successfully, something more along the lines of what I believe the OP wanted: "What do $\sigma$-fields have to do with probability theory?" or "Why is this author saying the 'foundations of probability theory' depend on $\sigma$-fields?"1

However, there's a very good chance that there is no reasonable reformulation because the question the OP wants to ask is just not a good question for Math.SE. The problem is the "guidance" part, especially when combined with the "broad" part. To provide a good answer to such a question requires knowing quite a bit about the background, what research they've already done, and the goals of the asker. The resulting answers tend to not be generalizable and are pretty opinion-based2.

Really, though, for myself the way these questions play out in practice is even more of a reason not to "welcome" them.

First, in my experience and in this particular case, these questions lack evidence of any research. If you put "foundations of probability theory" into Google with the quotes (let alone without them), the first hit is the Wikipedia page which, as far as I can tell, does a much better job of answering the OP's question than either of the given answers.3 The other hits are papers and lecture notes and blog posts and books on this topic. The story is similar for "sigma field". I'm pretty confident that the OP would have been completely satisfied with simply reading the Wikipedia pages on probability theory and sigma-algebras. I see zero reason to replicate such content on StackExchange.

Maybe the OP wouldn't have been satisfied or maybe the OP even did read those Wikipedia articles. This leads to second issue in practice in my experience. Extracting the background, prior research, and goals tends to become an exercise in twenty questions. I'm pretty sure this is frustrating all around.

Third, given the breadth of the topic and the vagueness of the target audience, the answers tend to be shallow and limited to a relatively narrow perspective. The shallowness is unsurprising but certainly not a virtue. The narrowness is a personal bugbear of mine: in this case it's the fact that no one felt any need to mention that there are, in fact, other "foundations to probability theory" that aren't based on $\sigma$-algebras.

In summary, Wikipedia provides better answers to these types of questions making answering them on StackExchange redundant at best. In the case where the OP has already consulted Wikipedia or some suitable analogue yet is still asking the question here, figuring out what exactly they're looking for is painful. (This is so because otherwise they would have already asked a much more narrow and targeted question that specified in what ways Wikipedia failed to satisfy them.) "Welcoming" such questions is to attempt to replicate Wikipedia poorly and/or to proliferate questions that primarily differ by the person asking them. For example, in this case the OP claimed to have "understood the definition of a $\sigma$-field". Is the same question asked by someone who instead claims not to understand the definition of a $\sigma$-field a different question? If not, why not? What if the person wants something more rigorous than the answers given to the current question? What if the person wants an answer geared to a data scientist instead of a linguist? If all of these should be treated as duplicates of the current question, then it is unambiguously too broad and the answers are quite inadequate.

In this particular case, I'd be fairly comfortable close voting this question based on either of "too broad" or "missing context". There's a touch of "primarily opinion-based" and/or "seeking personal advice" in there too but probably not so much that I'd close vote for those reasons.

1 These are certainly not ideal questions as stated.

2 Indeed, I would say the opinions of an expert are exactly the valuable thing being sought.

3 This is not to demean the answers. The Wikipedia article surely has orders of magnitude more hours put into it.

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    $\begingroup$ +1 even though we disagree. But I might think twice before I answer another one like this. Or perhaps answer it and vote to close. $\endgroup$ – Ethan Bolker Dec 17 '18 at 14:32
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    $\begingroup$ I've been pondering about this answer for some time now. It makes intuitive sense to me, however it does leave an open point of what type of questions then are suitable for Maths.SE. What is your vision on this @Derek? $\endgroup$ – Lord_Farin Dec 18 '18 at 17:36
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    $\begingroup$ I disagree with the Wikipedia argument. This forum allows several answers connected to one question and explicit ranking by votes, thereby implicitly recognizing that their is not necessarily one correct answer. However Wikipedia provides only one answer presumably aiming to given an authoritative one, providing little room for different points of view. Also Wikipedia aims to give encyclopedic answer at least judging by it's name. I do no believe in the possibility of making a distinction in questions suited for one forum versus the other ex ante hence the voting system. $\endgroup$ – Jesper Hybel Dec 24 '18 at 16:14
  • $\begingroup$ The first point made (that we shouldn't encourage this from new users) is an excellent one. I think it would typically take a good deal of experience on the site before deciding whether or not to ask a question like this. Apparently, though, a new user could strike upon such a question too, but I think that must be relatively rare. +1 $\endgroup$ – rschwieb Dec 24 '18 at 22:06
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It seems to me that for any area of mathematics there's a hierarchy of levels of understanding and of detail. And that a useful answer is one either at the same level as the question or one level deeper, but no more.

Seen this way, many "over-general" questions are in reality entirely specific: the questioner has one particular piece of knowledge they wish to acquire, and has specified clearly what it is.

An engineering example: someone might ask "How does Boolean algebra help in designing logic circuits?" Now there are big fat textbooks on that, and you could launch into explanations of Karnaugh maps, finite state machines and so on, but you'd not be answering the question. They want to know the basics: e.g. that there's a circuit element corresponding to each Boolean operation, that a Boolean expression translates directly into instructions for wiring such elements together, and that simplifying the expression also simplifies the circuit. You might also need to mention that $1$ and $0$ are usually represented by two different voltages.

If you went on to give an example expression and a corresponding circuit diagram, or to start doing some actual Boolean algebra, you'd be moving on to the next level, but the questioner would probably find it a helpful and interesting elaboration. If you went further though, you'd be outside the scope of the question and probably have lost them.

There may be questions that need an entire book—"How do I solve differential equations?"—but others would only need that if they were answered at an inappropriate level.

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  • $\begingroup$ "How does Boolean algebra help in designing logic circuits?" -1 as the example is not comparable. $\endgroup$ – The Great Duck Dec 28 '18 at 7:56

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