Borel–Kolmogorov paradox

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

A great circle puzzleEdit

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   uniformly from   and choosing the latitude   from   with density  .[1] Then we can look at two different great circles:

  1. If the coordinates are chosen so that the great circle is an equator (latitude  ), the conditional density for a longitude   defined on the interval   is
     
  2. If the great circle is a line of longitude with  , the conditional density for   on the interval   is
     

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.

Many quite futile arguments have raged — between otherwise competent probabilists — over which of these results is 'correct'.

Explanation and implicationsEdit

In case (1) above, the conditional probability that the longitude λ lies in a set E given that φ = 0 can be written P(λE | φ = 0). Elementary probability theory suggests this can be computed as P(λE 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 Rab = {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(φF | λ = 0) is defined using the events Lab = {λ : a < λ < b}, which are lunes (vertical wedges), consisting of all points whose longitude varies between a and b. So although P(λE | φ = 0) and P(φF | 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(λE | φ = 0) and P(φF | λ = 0) have different distributions.

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

… 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.

Mathematical explicationEdit

Using calculusEdit

Consider a random vector   that is uniformly distributed on the unit sphere  . The uniform distribution is defined in terms of surface integrals: for a given region  ,

 

This means that for any parametrization   that covers D, we have

 

For simplicity, we won't calculate the full conditional distribution on a great circle, only the probability that it lies in the first quadrant. That is to say, we will attempt to calculate the conditional probability   with

 

We begin by parametrizing the sphere with the usual spherical polar coordinates:

 

where   and  .

We can define random variables  ,   as the values of   under the inverse of this parametrization, or more formally using the arctan2 function:

 

Via the formula above, the probability distribution on the sphere is

 

This equation in effect defines the joint density   of   and  :

 .

We can rewrite A and B in polar coordinates:

 

We attempt to evaluate the conditional probability using the density

 

Now we repeat the process with a different parametrization of the sphere:

 

This is equivalent to the previous parametrization rotated by 90 degrees around the x axis.

Define new random variables

 

Rotation is measure preserving so the density of   and   is the same:

 .

The expressions for A and B are:

 

Attempting again to evaluate the conditional probability using the density

 

This shows that the conditional density cannot be treated as conditioning on an event of probability zero, as explained in Conditional probability#Conditioning on an event of probability zero.

Using measure theoryEdit

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

 

for which we obtain the volume element

 

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

 

Let

 

denote the joint measure on  , which has a density   with respect to   and let

 

If we assume that the density   is uniform, then

 

Hence,   has a uniform density with respect to   but not with respect to the Lebesgue measure. On the other hand,   has a uniform density with respect to   and the Lebesgue measure.

ReferencesEdit

CitationsEdit

  1. ^ a b c Jaynes 2003, pp. 1514–1517
  2. ^ Originally Kolmogorov (1933), translated in Kolmogorov (1956). Sourced from Pollard (2002)

SourcesEdit

  • Jaynes, E. T. (2003). "15.7 The Borel-Kolmogorov paradox". Probability Theory: The Logic of Science. Cambridge University Press. pp. 467–470. ISBN 0-521-59271-2. MR 1992316.
  • Kolmogorov, Andrey (1933). Grundbegriffe der Wahrscheinlichkeitsrechnung (in German). Berlin: Julius Springer.
  • Pollard, David (2002). "Chapter 5. Conditioning, Example 17.". A User's Guide to Measure Theoretic Probability. Cambridge University Press. pp. 122–123. ISBN 0-521-00289-3. MR 1873379.
  • Mosegaard, K., & Tarantola, A. (2002). 16 Probabilistic approach to inverse problems. International Geophysics, 81, 237–265.