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| In [[probability theory]], the '''Borel–Kolmogorov paradox''' (sometimes known as '''Borel's paradox''') is a [[paradox]] relating to [[conditional probability]] with respect to an [[event (probability theory)|event]] of probability zero (also known as a [[null set]]). It is named after [[Émile Borel]] and [[Andrey Kolmogorov]].
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| == A great circle puzzle ==
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| Suppose that a random variable has a [[Uniform distribution (continuous)|uniform distribution]] on a unit sphere. What is its [[conditional distribution]] on a [[great circle]]? Because of the symmetry of the sphere, one might expect that the distribution is uniform and independent of the choice of coordinates. However, two analyses give contradictory results. First, note that choosing a point uniformly on the sphere is equivalent to choosing the [[longitude]] ''λ'' uniformly from ''[-π,π]'' and choosing the [[latitude]] ''φ'' from ''[-π/2,π/2]'' with density <math>\frac{1}{2} \cos \phi </math>.<ref name=Jaynes>{{harvnb|Jaynes|2003|pages=1514–1517}}</ref> Then we can look at two different great circles:
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| :1. If the coordinates are chosen so that the great circle is an [[equator]] (latitude ''φ'' = 0), the conditional density for a longitude ''λ'' defined on the interval [–''π'',''π''] is
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| ::<math> f(\lambda|\phi=0) = \frac{1}{2\pi}.</math>
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| :2. If the great circle is a [[line of longitude]] with ''λ'' = 0, the conditional density for ''φ'' on the interval [–''π''/2,''π''/2] is
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| ::<math>f(\phi|\lambda=0) =\frac{1}{2} \cos \phi.</math>
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| One distribution is uniform on the circle, the other is not. Yet both seem to be referring to the same great circle in different coordinate systems.
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| {{Quote|
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| Many quite futile arguments have raged - between otherwise competent probabilists - over which of these results is 'correct'.
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| |[[E.T. Jaynes]]<ref name=Jaynes/>}}
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| == Explanation and implications ==
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| In case (1) above, the conditional probability that the longitude λ lies in a set ''E'' given that φ = 0 can be written ''P''(λ ∈ ''E'' | φ = 0). Elementary probability theory suggests this can be computed as ''P''(λ ∈ ''E'' and φ=0)/''P''(φ=0), but that expression is not well-defined since ''P''(φ=0) = 0. [[Measure theory]] provides a way to define a conditional probability, using the family of events ''R''<sub>''ab''</sub> = {φ : ''a'' < φ < ''b''} which are horizontal rings consisting of all points with latitude between ''a'' and ''b''.
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| The resolution of the paradox is to notice that in case (2), ''P''(φ ∈ ''F'' | λ=0) is defined using the events ''L''<sub>''ab''</sub> = {λ : ''a'' < λ < ''b''}, which are [[Lune (mathematics)#Spherical geometry|lunes]] (vertical wedges), consisting of all points whose longitude varies between ''a'' and ''b''. So although ''P''(λ ∈ ''E'' | φ=0) and ''P''(φ ∈ ''F'' | λ=0) each provide a probability distribution on a great circle, one of them is defined using rings, and the other using lunes. Thus it is not surprising after all that ''P''(λ ∈ ''E'' | φ=0) and ''P''(φ ∈ ''F'' | λ=0) have different distributions.
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| {{Quote|
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| The concept of a conditional probability with regard to an isolated hypothesis whose probability equals 0 is inadmissible. For we can obtain a probability distribution for [the latitude] on the meridian circle only if we regard this circle as an element of the decomposition of the entire spherical surface onto meridian circles with the given poles
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| |[[Andrey Kolmogorov]]<ref>Originally [[#kol1933|Kolmogorov (1933)]], translated in [[#kol1956|Kolmogorov (1956)]]. Sourced from [[#pol2002|Pollard (2002)]]</ref>}}
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| {{Quote|
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| … the term 'great circle' is ambiguous until we specify what limiting operation is to produce it. The intuitive symmetry argument presupposes the equatorial limit; yet one eating slices of an orange might presuppose the other.
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| |[[E.T. Jaynes]]<ref name=Jaynes/>}}
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| == Mathematical explication ==
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| To understand the problem we need to recognize that a distribution on a continuous random variable is described by a density ''f'' only with respect to some measure ''μ''. Both are important for the full description of the probability distribution. Or, equivalently, we need to fully define the space on which we want to define ''f''.
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| Let ''Φ'' and ''Λ'' denote two random variables taking values in ''Ω''<sub>''1''</sub>'' = [-π/2,π/2]'' respectively ''Ω''<sub>''2''</sub>'' = [-π,π]''. An event ''{Φ=φ,Λ=λ} '' gives a point on the sphere ''S(r)'' with radius ''r''. We define the [[coordinate transform]]
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| :<math>x = r \cos \phi \cos \lambda</math>
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| :<math>y = r \cos \phi \sin \lambda</math>
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| :<math>z = r \sin \phi</math>
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| for which we obtain the [[volume element]]
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| :<math>\omega_r(\phi,\lambda) = \left|\left| {\partial (x,y,z) \over \partial \phi} \times {\partial (x,y,z) \over \partial \lambda} \right|\right| = r^2 \cos \phi \ .</math>
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| Furthermore, if either ''φ'' or ''λ'' is fixed, we get the volume elements
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| :<math>\omega_r(\lambda) = \left|\left| {\partial (x,y,z) \over \partial \phi } \right|\right| = r \ , \quad\mathrm{respectively}</math>
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| :<math>\omega_r(\phi) = \left|\left| {\partial (x,y,z) \over \partial \lambda } \right|\right| = r \cos \phi\ .</math>
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| Let
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| :<math>\mu_{\Phi,\Lambda}(d\phi, d\lambda) = f_{\Phi,\Lambda}(\phi,\lambda) \omega_r(\phi,\lambda) d\phi d\lambda</math>
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| denote the joint measure on <math>\mathcal{B}(\Omega_1 \times \Omega_2)</math>, which has a density <math>f_{\Phi,\Lambda}</math> with respect to <math>\omega_r(\phi,\lambda) d\phi d\lambda</math> and let
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| :<math>\mu_{\Phi}(d\phi) = \int_{\lambda \in \Omega_2} \mu_{\Phi,\Lambda}(d\phi, d\lambda)\ ,</math>
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| :<math>\mu_{\Lambda}(d\lambda) = \int_{\phi \in \Omega_1} \mu_{\Phi,\Lambda}(d\phi, d\lambda)\ .</math>
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| If we assume that the density <math>f_{\Phi,\Lambda}</math> is uniform, then
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| :<math>\mu_{\Phi | \Lambda}(d\phi | \lambda) = {\mu_{\Phi,\Lambda}(d\phi, d\lambda) \over \mu_{\Lambda}(d\lambda)} = \frac{1}{2r} \omega_r(\phi) d\phi \ , \quad\mathrm{and}</math>
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| :<math>\mu_{\Lambda | \Phi}(d\lambda | \phi) = {\mu_{\Phi,\Lambda}(d\phi, d\lambda) \over \mu_{\Phi}(d\phi)} = \frac{1}{2r\pi} \omega_r(\lambda) d\lambda \ .</math>
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| Hence, <math>\mu_{\Phi | \Lambda}</math> has a uniform density with respect to <math>\omega_r(\phi) d\phi</math> but not with respect to the Lebesgue measure. On the other hand, <math>\mu_{\Lambda | \Phi}</math> has a uniform density with respect to <math>\omega_r(\lambda) d\lambda</math> and the Lebesgue measure.
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| == Notes ==
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| {{Reflist}}
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| ==References and further reading==
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| {{refbegin}}
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| * {{cite book |title= Probability Theory: The Logic of Science |last= Jaynes |first= E.T. |authorlink= Edwin Thompson Jaynes |coauthors= |year= 2003 |publisher= Cambridge University Press |isbn= 0-521-59271-2 |page= |pages= 467–470 |url= |chapter= 15.7 The Borel-Kolmogorov paradox |ref= jay2003 |mr=1992316 }}
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| ** [http://omega.math.albany.edu:8008/ETJ-PS/cc15w.ps Fragmentary Edition (1994) (pp. 1514–1517)] ([[PostScript]] format)
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| * {{cite book |title= Grundbegriffe der Wahrscheinlichkeitsrechnung |last= Kolmogorov |first= Andrey |authorlink= Andrey Kolmogorov |coauthors= |year= 1933 |publisher= Julius Springer |location= Berlin |isbn= |page= |pages= |url= |language= German |ref= kol1933}}
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| ** Translation: {{cite book |title= Foundations of the Theory of Probability |edition= 2nd |last= Kolmogorov |first= Andrey |authorlink= Andrey Kolmogorov |coauthors= |year= 1956 |publisher= Chelsea |location= New York |isbn= 0-8284-0023-7 |page= |pages= 50–51|url= http://www.mathematik.com/Kolmogorov/index.html |ref= kol1956 |chapter= Chapter V, §2. Explanation of a Borel Paradox| chapterurl= http://www.mathematik.com/Kolmogorov/0029.html}}
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| * {{cite book |title= A User's Guide to Measure Theoretic Probability |last= Pollard |first= David |coauthors= |year= 2002 |publisher= Cambridge University Press|isbn= 0-521-00289-3 |page= |pages=122–123 |url= |chapter= Chapter 5. Conditioning, Example 17. |ref= pol2002 |mr=1873379 }}
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| {{refend}}
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| * Mosegaard, K., & Tarantola, A. (2002). 16 Probabilistic approach to inverse problems. International Geophysics, 81, 237-265.
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| {{DEFAULTSORT:Borel-Kolmogorov Paradox}}
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| [[Category:Probability theory paradoxes]]
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