Timeline for Is there an introduction to probability theory from a structuralist/categorical perspective?
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| Feb 8, 2023 at 20:26 | history | edited | Mark Meckes | CC BY-SA 4.0 |
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| Jul 8, 2018 at 9:51 | comment | added | Watson | "I was surprised by Kazhdan’s request since “everybody knows” that a random variable is just a measurable function $X(ω)$ from $Ω$ to $\mathcal X$. He answered “yes, but that’s not what it means to people working in probability” and of course he was right." — P. Diaconis | |
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Mar 26, 2015 at 10:42 | comment | added | Mark Meckes | @MartinBrandenburg: Moreover, I would need convincing that probability theory is about the category of probability spaces and measurable maps. That category surely plays a role, but most of what's of interest in the theory seems to take place outside of it. In particular, if you can show me how to view the law of large numbers or the central limit theorem as being about the category of probability spaces and measurable maps, then I'll reconsider. | |
| Mar 26, 2015 at 1:59 | comment | added | Mark Meckes | @MartinBrandenburg: I don't claim that it's distracting to define graphs as pairs of sets. I claim that if you do get distracted by the details of such a definition, then you're not doing graph theory. | |
| Mar 25, 2015 at 22:56 | comment | added | Martin Brandenburg | One can talk about a field without having to write it down as a tuple. Just say $K$ is the field, an object of the category of fields and not of the category of sets, and don't confuse it with its underlying set $|K|$. Similarly, we can talk about graphs without having to think of them as pairs of sets. The answer claims that it is somehow difficult or distracting to define graphs that way, but I don't agree with this either. | |
| Mar 25, 2015 at 22:53 | comment | added | Martin Brandenburg | By the way, I disagree with Rudin's comment. It basically says "It is a tradition to ignore forgetful functors, so I will follow this tradition" and "argues" for this procedure by saying that otherwise one would "have to" imagine the reals as a quadruple. No. It is really about the question in which category one works. There is a field of real numbers, there is a measure space of real numbers, there is a topological space of real numbers, etc., and it is very unfortunate that all are denoted by the same symbol. At least, one should not forget the forgetful functors between these categories. | |
| Mar 25, 2015 at 22:47 | comment | added | Martin Brandenburg | "probability theory is not about probability spaces" --- yes, but it is perhaps about the category of probability spaces and measurable maps. Random variables are morphisms in that category. The approach to look at families of random variables to understand some distribution is really in the spirit of the Yoneda Lemma. Morphisms are more important than objects. | |
| Nov 20, 2013 at 3:25 | comment | added | Daniel McLaury | Re: the quote from Rudin, I once graded a real analysis course using Wade's text, which essentially proceeds by saying "assume there exists a complete ordered field" and then proving theorems that hold in such an object. I had to tweak the way I graded the course since students didn't really appreciate that perspective on analysis at that stage in their development. | |
| Sep 23, 2012 at 20:27 | comment | added | Mark Meckes | @Vectornaut: No, it doesn't. For a start, this equivalence doesn't mention anything about a measure on the one side or an expectation on the other side. Maybe the equivalence can be made to include those data as well, but the major issue -- if you want to think about things in such terms -- is that in probability theory you should not only mod out by sets of measure 0, you should also mod out by measure-automorphisms of the domain which preserve the joint distribution of the random variables under consideration. | |
| Sep 23, 2012 at 16:33 | comment | added | Vectornaut | Dmitri Pavlov's answer mentions that "The category of localizable measurable spaces is equivalent to the category of commutative von Neumann algebras." Doesn't this mean that studying families of random variables is the same thing as studying probability spaces? | |
| Sep 3, 2010 at 19:08 | comment | added | Terry Tao | Just for the record, that quote is my own, though the general sentiment that probability is not about measure spaces is certainly very widely held among probabilists. | |
| Aug 23, 2010 at 20:39 | comment | added | Nate Eldredge | The quote I mentioned above appears in Terry Tao's blog at terrytao.wordpress.com/2010/01/01/… dated January 1, 2010, which is cited in another answer to this question. I am not sure if that is where I read it, but it's the earliest reference I have so far. | |
| Jun 1, 2010 at 15:12 | comment | added | Mark Meckes | @Carl: Indeed! The paragraph following the one you pointed out is even better for getting to the real point here. For anyone else interested, here's the link to Fremlin's treatise (although it's easy enough to find via Google): essex.ac.uk/maths/staff/fremlin/mt.htm Watch out for the fact that it's plain TeX (not LaTeX)! | |
| May 31, 2010 at 13:46 | comment | added | Carl Offner | Fair enough. If you get a chance, take a look at Fremlin's statement. I think you'll like it. | |
| May 31, 2010 at 3:16 | comment | added | Mark Meckes | You can't talk about expectations if all you have is a probability space. You need to look at a measurable function (random variable) from your probability space into $\mathbb{R}$ or a similar algebraic structure; or equivalently you need your probability space itself to have some algebraic structure. | |
| May 31, 2010 at 2:24 | comment | added | Carl Offner | I'm not sure exactly what you mean by additional structure, because it seems to me that expectations are pretty immediate. But I think your general point is correct. A great quote illuminating that (which due to space limitations I can't quite copy here) is in Fremlin's Measure Theory treatise -- Vol.2, the second paragraph of the introduction to Chapter 27. I think it's well worth looking at, and also very well written | |
| May 30, 2010 at 23:35 | comment | added | Mark Meckes | That is a great quote, but it doesn't make all the points Nate's quote does. If you think of the reals as a quadruple, you have the formalism necessary to understand and prove theorems about real numbers, although you may lack the intuition needed to appreciate the theorems. But if you think of natural numbers as strings of digits, you're missing not only intuition but also interesting algebraic structure. Likewise, a measure space with measure one is insufficient structure for probability; you need some additional algebraic or geometric structure before you can even talk about expectations. | |
| May 30, 2010 at 14:47 | comment | added | Carl Offner | Another great quote along the same lines, from Rudin (Real and Complex Analysis, page 18 in my edition): "For instance, the real line may be described as a quadruple $(R^1, +, \cdot, <)$ where $+$, $\cdot$, and $<$ satisfy the axioms of a complete archimedean ordered field. But it is a safe bet that very few mathematicians think of the real field as an ordered quadruple." | |
| May 26, 2010 at 11:18 | comment | added | Mark Meckes | Yes, I've seen that quote too, and it's better than my analogies now that you remind me of it. But I also forget the author. | |
| May 25, 2010 at 13:36 | comment | added | Nate Eldredge | Indeed, I saw a quote from somebody famous (if I think of the author I'll edit) to the effect that "one could say that probability theory is the study of measure spaces with measure one, but this is like saying that number theory is the study of finite strings of the digits {0,...,9}." | |
| Apr 12, 2010 at 13:38 | history | edited | Mark Meckes | CC BY-SA 2.5 |
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| Apr 9, 2010 at 12:28 | history | answered | Mark Meckes | CC BY-SA 2.5 |