Talk:Borel–Kolmogorov paradox

images and examples

I have created two images, Image:Borelparardox support xydist.png and Image:Borelparardox support uvdist.png, showing the regions on which the two distributions are non-zero. I'm not handy enough at image manipulation and html to insert them into the document, but they're available to anyone who wishes to do so and has the know-how. -- Cyan 21:55, 26 Jan 2004 (UTC)

Credit where credit is due: the example of the paradox that I wrote up was taken, mutatis mutandis, from [1]. -- Cyan 04:06, 27 Jan 2004 (UTC)

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? Angela. 17:14, Jan 27, 2004 (UTC)

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! -- Cyan 17:38, 27 Jan 2004 (UTC)

The links to the images don't work. The village pump says: "Note that images uploaded from Jan 24-Jan 28 are unavailable for now; try re-uploading anything you lost." Fpahl 12:33, 5 Apr 2004 (UTC)

another explanation

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 coordinate systems. But I'm not sure how much consensus there is on this view of the paradox -- any comments? Fpahl 18:25, 19 Apr 2004 (UTC)

It's the explanation I endorse, although others exist. Go for it. -- Cyan

too technical tag

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. Lunch 04:34, 24 September 2006 (UTC)

Dificult to understand

I wouldn't even say this is too technical. It is just badly defined: An event {Φ=φ,Λ=λ} is a point on the sphere S(r) with radius r. — Preceding unsigned comment added by André Caldas (talk • contribs) 12:38, 21 September 2012 (UTC)

example

I think the sphere coordinates example mentioned in this sci.math thread is more illustrative than the current example.--Novwik (talk) 19:15, 17 November 2007 (UTC)

I agree. I've started to rewrite the article around the spherical example. I also think the article should be Borel-Kolmogorov paradox, as that seems to be the most common reference. -3mta3 (talk) 10:31, 10 March 2009 (UTC)

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 -3mta3 (talk) 12:19, 11 March 2009 (UTC)

reconsider canonical example wording

The article says:

"Consider a random point distributed uniformly over the surface of an assumed spherical "earth""

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." —Preceding unsigned comment added by 113.190.136.203 (talk) 15:49, 8 August 2010 (UTC)

Uniform distribution

I reverted most of the edits by Pfbenner. The initial explanation of the paradox got too confused with notation, and more importantly it became incorrect: it's not a paradox if you don't explain that the point is uniformly chosen on the sphere. The way it was written it sounded like the latitude and longitude were chose uniformly, seperately. --Dylan Thurston (talk) 14:58, 21 October 2012 (UTC)

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 this paradox. That's why I've chosen a more technical description and added the mathematical explication. Philipp Benner (talk) 18:18, 21 October 2012 (UTC)

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 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. --Dylan Thurston (talk) 20:47, 21 October 2012 (UTC)

I changed it to be “uniformly distributed with respect to surface area”, which I think is unambiguous. ---Dylan Thurston (talk) 21:31, 21 October 2012 (UTC)

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. Philipp Benner (talk) 08:29, 27 October 2012 (UTC)

Limiting operation example

I removed the following text

One natural way to choose Jaynes' "limiting operation" is by specifying a notion of distance on the space. For example, if we choose the usual Euclidean distance, we obtain a uniform distribution on great circles. However, choosing a different notion of distance would result in the $\cos \phi /2$ 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 Hausdorff measure.

There's no citation, so it appears to be original research, 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.

Azaghal of Belegost (talk) 21:59, 22 February 2019 (UTC)

Error in "Mathematical Explication"

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 the second parameterisation,

$\mathbb {P} (A\cup A_{4}\mid B){\stackrel {?}{=}}\int _{-{\frac {\pi }{2}}}^{\frac {\pi }{2}}f_{\Phi '|\Theta '}(\varphi ,0)\mathrm {d} \varphi =\int _{-{\frac {\pi }{2}}}^{\frac {\pi }{2}}{\frac {1}{2}}\cos(\varphi )\mathrm {d} \varphi =1.$

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 $\Theta ',\Theta ,\theta$.

Providing an erroneous illustration to a hard-to-grasp topic is worse than providing none.

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. AVM2019 (talk) 01:57, 26 March 2021 (UTC)

I added the "Original Research" plaque to warn of possible unreliable content. The contribution is in question is this one by User:User357269. AVM2019 (talk) 21:34, 26 March 2021 (UTC)

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. User357269 (talk) 08:25, 27 March 2021 (UTC)

I suggest you remove the original research: it does not belong in Wikipedia even if you "fix it". Regards, AVM2019 (talk) 17:11, 28 March 2021 (UTC)

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. User357269 (talk) 22:59, 28 March 2021 (UTC)

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. 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 Regular conditional probability 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. AVM2019 (talk) 00:12, 29 March 2021 (UTC)

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. User357269 (talk) 00:33, 29 March 2021 (UTC)

Not really puzzling

It may be difficult to point out exactly where the erroneous argument goes wrong.

But at least to me it is immediately obvious 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 conditional distribution must also be uniform on C. 2601:200:C000:1A0:8471:14C6:2E4:8561 (talk) 00:36, 15 May 2022 (UTC)

We use the word paradox precisely when an immediately obvious thing turns out to be deceiving. AVM2019 (talk) 01:11, 17 May 2022 (UTC)

I am well aware of that usage of the word "paradox", but what I wrote above is true. 2601:200:C000:1A0:7D2E:86D:EFAE:755 (talk) 19:40, 21 May 2022 (UTC)

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! AVM2019 (talk) 23:21, 22 May 2022 (UTC)

I have not read Borel or Kolmogorov's original.

But I am not "suggesting" anything. I am stating a fact:

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.

If the Wikipedia article contradicts this fact, then it is overwhelmingly more likely that a Wikipedia editor made a mistake than that Borel or Kolmoogorov did. 2601:200:C000:1A0:C820:3D3D:D3F2:D688 (talk) 23:59, 24 May 2022 (UTC)

There is also an option that neither Borel and Kolmogorov, nor Wikipedia authors were mistaken, but rather the person stating the "fact", which, in fact, is not a fact. AVM2019 (talk) 23:27, 28 May 2022 (UTC)

I am curious to learn what your background in mathematics is. 2601:200:C000:1A0:2CF1:CC57:B2A0:8D4E (talk) 15:37, 8 June 2022 (UTC)

The fact that I stated above is immediately obvious to any mathematician. That is why I asked. 2601:200:C000:1A0:D17A:DAB0:4CFB:E335 (talk) 17:42, 9 June 2022 (UTC)

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 you.

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 learn to be cautious.

I see no reason in continuing this conversation, unless you point at something specific in the article what you think is a mistake and make a valid argument. I suggest you read any of the references explaining the paradox. I personally find Jaynes' 2003 book easy to follow. AVM2019 (talk) 01:46, 10 June 2022 (UTC)

Have you actually looked at what either Borel or Kolmogorov wrote?

Because you seem to be assuming that the article is an accurate representation of the paradox.

It is not.

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 "point at something specific in the article [that I] think is a mistake and make a valid argument". 2601:200:C000:1A0:9D6A:3426:156B:13FB (talk) 00:31, 17 June 2022 (UTC)

Sorry, I do not see where you were being specific, but I see you admitting the contrary in your first message. All the best AVM2019 (talk) 19:50, 20 June 2022 (UTC)