Talk:Borel–Kolmogorov paradox

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images and examples[edit]

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[edit]

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[edit]

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[edit]

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 (talkcontribs) 12:38, 21 September 2012 (UTC)

example[edit]

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[edit]

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[edit]

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)