Finite impulse response

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 of probability zero (also known as a null set). It is named after Émile Borel and Andrey Kolmogorov.

A great circle puzzle

Suppose that a random variable has a 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 ${\frac {1}{2}}\cos \phi$. Then we can look at two different great circles:

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

$f(\lambda |\phi =0)={\frac {1}{2\pi }}.$

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

$f(\phi |\lambda =0)={\frac {1}{2}}\cos \phi .$

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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Explanation and implications

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_ab = {φ : a < φ < b} which are horizontal rings consisting of all points with latitude between a and b.

The resolution of the paradox is to notice that in case (2), P(φ ∈ F | λ=0) is defined using the events L_ab = {λ : a < λ < b}, which are 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.

31 year-old Systems Analyst Bud from Deep River, spends time with pursuits for instance r/c cars, property developers new condo in singapore singapore and books. Last month just traveled to Orkhon Valley Cultural Landscape. 31 year-old Systems Analyst Bud from Deep River, spends time with pursuits for instance r/c cars, property developers new condo in singapore singapore and books. Last month just traveled to Orkhon Valley Cultural Landscape.

Mathematical explication

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.

Let Φ and Λ denote two random variables taking values in Ω₁ = [-π/2,π/2] respectively Ω₂ = [-π,π]. An event {Φ=φ,Λ=λ} gives a point on the sphere S(r) with radius r. We define the coordinate transform

$x=r\cos \phi \cos \lambda$

$y=r\cos \phi \sin \lambda$

$z=r\sin \phi$

for which we obtain the volume element

$\omega _{r}(\phi ,\lambda )=\left|\left|{\partial (x,y,z) \over \partial \phi }\times {\partial (x,y,z) \over \partial \lambda }\right|\right|=r^{2}\cos \phi \ .$

Furthermore, if either φ or λ is fixed, we get the volume elements

$\omega _{r}(\lambda )=\left|\left|{\partial (x,y,z) \over \partial \phi }\right|\right|=r\ ,\quad \mathrm {respectively}$

$\omega _{r}(\phi )=\left|\left|{\partial (x,y,z) \over \partial \lambda }\right|\right|=r\cos \phi \ .$

Let

$\mu _{\Phi ,\Lambda }(d\phi ,d\lambda )=f_{\Phi ,\Lambda }(\phi ,\lambda )\omega _{r}(\phi ,\lambda )d\phi d\lambda$

denote the joint measure on ${\mathcal {B}}(\Omega _{1}\times \Omega _{2})$, which has a density $f_{\Phi ,\Lambda }$ with respect to $\omega _{r}(\phi ,\lambda )d\phi d\lambda$ and let

$\mu _{\Phi }(d\phi )=\int _{\lambda \in \Omega _{2}}\mu _{\Phi ,\Lambda }(d\phi ,d\lambda )\ ,$

$\mu _{\Lambda }(d\lambda )=\int _{\phi \in \Omega _{1}}\mu _{\Phi ,\Lambda }(d\phi ,d\lambda )\ .$

If we assume that the density $f_{\Phi ,\Lambda }$ is uniform, then

$\mu _{\Phi |\Lambda }(d\phi |\lambda )={\mu _{\Phi ,\Lambda }(d\phi ,d\lambda ) \over \mu _{\Lambda }(d\lambda )}={\frac {1}{2r}}\omega _{r}(\phi )d\phi \ ,\quad \mathrm {and}$

$\mu _{\Lambda |\Phi }(d\lambda |\phi )={\mu _{\Phi ,\Lambda }(d\phi ,d\lambda ) \over \mu _{\Phi }(d\phi )}={\frac {1}{2r\pi }}\omega _{r}(\lambda )d\lambda \ .$

Hence, $\mu _{\Phi |\Lambda }$ has a uniform density with respect to $\omega _{r}(\phi )d\phi$ but not with respect to the Lebesgue measure. On the other hand, $\mu _{\Lambda |\Phi }$ has a uniform density with respect to $\omega _{r}(\lambda )d\lambda$ and the Lebesgue measure.

Notes

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References and further reading

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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  • Fragmentary Edition (1994) (pp. 1514–1517) (PostScript format)
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  • Translation: 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
    
    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

• Mosegaard, K., & Tarantola, A. (2002). 16 Probabilistic approach to inverse problems. International Geophysics, 81, 237-265.