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distribution of Φ and Λ”, can be interpreted correctly: at no point did you mention surface area. You're also drawing some distinction between “uniform density” and “uniform distribution” that I fail to understand. The uniform distribution is the one whose density is constant with respect to surface area, in the use of the terms I'm used to. This is, of course, not the one that is uniform with respect to
Lebesgue measure in the λ and φ variables, as I think I explained. I don't mind changing “uniform distribution” to “uniform density with respect to surface area”, but I don't understand why you prefer the longer phrase. --
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668:, and incorrect as far as I can tell. A notion of distance is already present in order to define a uniform distribution over the sphere. The proposal to use Euclidean distance is presumably a proposal to say the distance is Euclidean in coordinates of latitude and longitude. A uniform distribution with respect to this distance would yield a distribution over the sphere that is decidedly not uniform in the sense discussed at the top of the article.
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Something does not add up in the example with two parameterisations. If instead of probability of finding the vector in the 1st quadrant (conditioned on it being on the great circle) we compute the probability of finding it in the 1st or 4th quadrants (conditioned on the same), we will obtain, using
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You are mistaken about your claim being obvious to "any mathematician". A mathematician would not use "immediately obvious" as an argument. Instead, they would try to come up with a formal and concrete argument. A mathematician would haven seen many paradoxes or have fallen for enough fallacies to
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First, your "simple calculation" turns out not so simple because you made a mistake in it. Second, it is still original research: you made this "proof of contradiction" up entirely by yourself. If not, please provide a reference. Third, details are not always useful, especially if they are flawed.
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Many of the more technical details you moved up were just irrelevant; eg, there is no need to specify the radius of the sphere, or to name it at all. But you also miss the important technical details. I cannot see how the description you wrote of the distribution, “a uniform density to the joint
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Please revert this or correct what you found confusing. Jaynes writes: "Given a uniform probability density over the surface area" and that's precisely what is wrong in the original description on this page. We choose a uniform density and not a uniform distribution, which is indeed the essence of
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I'm thinking of adding something about the problem being the artificial precision of an exact condition. The "paradox" does not appear if you specify a small but finite interval as a condition; it relies on the fact that the point-like condition has different relative "precision" in the two
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that any valid definition of "conditional distribution" will conclude that if the uniform distribution on the unit sphere is conditioned on a particular circle C of positive radius on the sphere, then the rotational symmetry of the original distribution about the axis of C implies that the
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It's more about how you think about the problem. The paradox usually appears when you do calculations with densities and don't think about possible interpretations. Once you define distributions and think about the probabilistic (or measure theoretic) interpretation the paradox is solved.
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Considering the size of the diagrams, I've just linked to them rather than displaying them inline, but I'm not really sure whether that is the best option. Do you think it would be more helpful to display on the page, and if so, do you want them in the positions the links currently are?
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I originally imagined them shrunken and inline, but I wasn't sure how to go about it. It didn't occur to me that they could be linked within the article. I am satisfied with the present arrangement; I will rely on the wiki process for improvements. Thanks, Angela! --
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That is, the random vector is almost never in the 2nd or 3rd quadrants. Even though this "conditional probability" is not properly defined, this breaking of symmetry is very surprising, and I strongly suspect an error, probably as a result of a confusion between
905:{\displaystyle \mathbb {P} (A\cup A_{4}\mid B){\stackrel {?}{=}}\int _{-{\frac {\pi }{2}}}^{\frac {\pi }{2}}f_{\Phi '|\Theta '}(\varphi ,0)\mathrm {d} \varphi =\int _{-{\frac {\pi }{2}}}^{\frac {\pi }{2}}{\frac {1}{2}}\cos(\varphi )\mathrm {d} \varphi =1.}
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It's now fixed. This isn't original research, it's a write-up of a simple calculation. I added the section because I think there is value in seeing the contradiction explicitly. Like you said, it's a hard-to-grasp topic, so details are useful.
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I don't have to invoke my own track record when I can simply appeal to the authority of the big names in the title and in the references. Because of this, the onus of proving that something is wrong is on
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You may think that your explanation is correct and clear, but it can really be wrong and confusing. I believe the latter is the case. By the way, I came across your harmful contribution to
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Could someone please find and correct the mistake. Or explain why the (ill-defined) "conditional probability" shows that the random vector is almost never in the 2nd or 3rd quadrants.
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If the uniform distribution on the unit sphere in 3-dimensional space is conditioned on a (not necessarily great) circle, the conditional distribution on that circle will be uniform.
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i removed the tag. if you wish to reinsert it, please do so but leave some suggestions here as to what could be improved or what you find difficult to understand. thanks.
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however, this doesn't seem to make sense. How do you distribute 1 point, let alone 1 anything? It's like saying "imagine 1 orange distributed uniformly around your house."
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Okay, I've rewritten most of it, and tried to emphasise the important concepts as I saw it. I might try and draw some pictures to aid the explanation if I get around to it -
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There is also an option that neither Borel and
Kolmogorov, nor Knowledge authors were mistaken, but rather the person stating the "fact", which, in fact, is not a fact.
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distribution. In general, once a notion of distance is specified, the natural choice for conditional distributions can generally be given in terms of the corresponding
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If the
Knowledge article contradicts this fact, then it is overwhelmingly more likely that a Knowledge editor made a mistake than that Borel or Kolmoogorov did.
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Are you suggesting that Borel and
Kolmogorov were wrong? I am sure that finding a mistake in their work and telling the world about it would be worth your time!
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completely independently, it just struck my eye. Finally, I insist that the old and verified content comes first in article. Please do not start an edit war.
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Thanks for pointing this out, this is 100% my mistake. The first quadrant probability is actually 1/4, so there's no paradox... I will try to fix it today.
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I agree with all your points and I'm sorry for making so many mistakes. If you have any suggestions on how I can improve the article, please let me know.
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I am not questioning the paradox. But the article misstates it. And if you were trained in mathematics, you would recognize that I already did "
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Sorry, I do not see where you were being specific, but I see you admitting the contrary in your first message. All the best
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I added the "Original
Research" plaque to warn of possible unreliable content. The contribution is in question is
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says: "Note that images uploaded from Jan 24-Jan 28 are unavailable for now; try re-uploading anything you lost."
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I agree. I've started to rewrite the article around the spherical example. I also think the article should be
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coordinate systems. But I'm not sure how much consensus there is on this view of the paradox -- any comments?
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I suggest you remove the original research: it does not belong in
Knowledge even if you "fix it". Regards,
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I changed it to be “uniformly distributed with respect to surface area”, which I think is unambiguous. ---
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this paradox. That's why I've chosen a more technical description and added the mathematical explication.
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Knowledge. If you would like to participate, please visit the project page, where you can join
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The fact that I stated above is immediately obvious to any mathematician. That is why I asked.
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Because you seem to be assuming that the article is an accurate representation of the paradox.
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Providing an erroneous illustration to a hard-to-grasp topic is worse than providing none.
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I am well aware of that usage of the word "paradox", but what I wrote above is true.
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One natural way to choose Jaynes' "limiting operation" is by specifying a notion of
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It may be difficult to point out exactly where the erroneous argument goes wrong.
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Credit where credit is due: the example of the paradox that I wrote up was taken,
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I wouldn't even say this is too technical. It is just badly defined: An event
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It's the explanation I endorse, although others exist. Go for it. --
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Have you actually looked at what either Borel or
Kolmogorov wrote?
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I am curious to learn what your background in mathematics is.
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I see no reason in continuing this conversation, unless you
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I think the sphere coordinates example mentioned in this
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But I am not "suggesting" anything. I am stating a fact:
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conditional distribution must also be uniform on C.
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406:André Caldas
400:— Preceding
394:
393:with radius
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335:village pump
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137:Mid-priority
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62:Mid‑priority
40:WikiProjects
500:—Preceding
261:peer review
259:received a
112:Mathematics
103:mathematics
59:Mathematics
1448:Categories
1388:It is not.
1122:User357269
1053:User357269
1004:User357269
387:{Φ=φ,Λ=λ}
201:Statistics
192:statistics
164:Statistics
977:this one
616:distance
526:Pfbenner
502:unsigned
414:contribs
402:unsigned
266:archived
1415:AVM2019
1370:AVM2019
1290:AVM2019
1211:AVM2019
1183:AVM2019
1175:paradox
1088:AVM2019
1026:AVM2019
986:AVM2019
961:AVM2019
425:example
302:, from
228:on the
139:on the
30:B-class
435:Novwik
316:Angela
36:scale.
472:3mta3
454:3mta3
370:Lunch
352:Fpahl
339:Fpahl
305:. --
1419:talk
1405:talk
1374:talk
1353:you.
1344:talk
1308:talk
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410:talk
391:S(r)
359:Cyan
328:Cyan
307:Cyan
293:Cyan
287:and
1399:".
980:by
874:cos
630:cos
397:.
220:Mid
131:Mid
1450::
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925:Θ
900:1.
894:φ
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855:π
844:π
839:−
835:∫
828:φ
811:φ
799:Θ
786:Φ
771:π
760:π
755:−
751:∫
725:∣
712:∪
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636:ϕ
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319:.
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42::
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