Useful vague question closed?

Question

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.

Answer 1

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.

Answer 2

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 reference-request, 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.

Answer 3

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.