Borel's paradox

Borel's paradox

Borel's paradox (sometimes known as the Borel-Kolmogorov paradox) is a paradox of probability theory relating to conditional probability density functions. The paradox lies in fact that, contrary to intuition, conditional probability density functions are not invariant under coordinate transformations.

Suppose we have two random variables, "X" and "Y", with joint probability density "pX,Y"("x","y"). We can form the conditional density for "Y" given "X",

:p_{Y|X}(y|x) = frac{p_{X,Y}(x,y)}{p_{X}(x)}

where "pX"("x") is the appropriate marginal distribution.

Using the substitution rule, we can reparametrize the joint distribution with the functions "U"= "f"("X","Y"), "V" = "g"("X","Y"), and can then form the conditional density for "V" given "U".

:p_{V|U}(v|u) = frac{p_{U,V}(u,v)}{p_{U}(u)}

Given a particular condition on "X" and the equivalent condition on "U", intuition suggests that the conditional densities "pY|X"("y"|"x") and "pV|U"("v"|"u") should also be equivalent. This is not the case in general.

A concrete example

A uniform distribution

We are given the joint probability density

:p_{X,Y}(x,y) =left{egin{matrix} 1, & 0 < y < 1, quad -y < x < 1 - y \ 0, & mbox{otherwise} end{matrix} ight.

The marginal density of "X" is calculated to be

:p_X(x) =left{egin{matrix} 1+x, & -1 < x le 0 \ 1 - x, & 0 < x < 1 \ 0, & mbox{otherwise}end{matrix} ight.

So the conditional density of "Y" given "X" is

:p_{Y|X}(y|x) =left{egin{matrix} frac{1}{1+x}, & -1 < x le 0, quad -x < y < 1 \ \ frac{1}{1-x}, & 0 < x < 1, quad 0 < y < 1 - x \ \ 0, & mbox{otherwise}end{matrix} ight.

which is uniform with respect to "y".

Reparametrization

Now, we apply the following transformation:

:U = frac{X}{Y} + 1 qquad qquad V = Y.

Using the substitution rule, we obtain

:p_{U,V}(u,v) =left{egin{matrix} v, & 0 < v < 1, quad 0 < u cdot v < 1 \ 0, & mbox{otherwise} end{matrix} ight.

The marginal distribution is calculated to be

:p_U(u) =left{egin{matrix} frac{1}{2}, & 0 < u le 1 \ \ frac {1}{2u^2}, & 1 < u < +infty \ \ 0, & mbox{otherwise}end{matrix} ight.

So the conditional density of "V" given "U" is

:p_{V|U}(v|u) =left{egin{matrix} 2v, & 0 < u le 1, quad 0 < v < 1 \ 2u^2v, & 1 < u < +infty, quad 0 < v < frac{1}{u} \ 0, & mbox{otherwise}end{matrix} ight.

which is not uniform with respect to "v".

The unintuitive result

Now we pick a particular condition to demonstrate Borel's paradox. The conditional density of "Y" given "X" = 0 is

:p_{Y|X}(y|x=0) = left{egin{matrix} 1, & 0 < y < 1 \ 0, & mbox{otherwise}end{matrix} ight.

The equivalent condition in the "u"-"v" coordinate system is "U" = 1, and the conditional density of "V" given "U" = 1 is

:p_{V|U}(v|u=1) = left{egin{matrix} 2v, & 0 < v < 1 \ 0, & mbox{otherwise}end{matrix} ight.

Paradoxically, "V" = "Y" and "X" = 0 is equivalent to "U" = 1, but

:p_{Y|X}(y|x = 0) e p_{V|U}(v|u = 1).;

ee also

* Émile Borel

References

*Jaynes, E. T., 2003, "Probability Theory: The Logic of Science", Cambridge University Press.


Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

  • Émile Borel — Infobox Person name = Félix Édouard Justin Émile Borel image size = 200px caption = Émile Borel birth date = birth date|1871|1|7|mf=y birth place = Saint Affrique, France death date = death date and age|1956|2|3|1871|1|7|mf=y death place = Paris …   Wikipedia

  • Infinity-Borel set — In set theory, a subset of a Polish space X is infin; Borel if itcan be obtained by starting with the open subsets of X, and transfinitely iterating the operations of complementation and wellordered union (but see the caveat below). Formal… …   Wikipedia

  • List of mathematics articles (B) — NOTOC B B spline B* algebra B* search algorithm B,C,K,W system BA model Ba space Babuška Lax Milgram theorem Baby Monster group Baby step giant step Babylonian mathematics Babylonian numerals Bach tensor Bach s algorithm Bachmann–Howard ordinal… …   Wikipedia

  • Conditioning (probability) — Beliefs depend on the available information. This idea is formalized in probability theory by conditioning. Conditional probabilities, conditional expectations and conditional distributions are treated on three levels: discrete probabilities,… …   Wikipedia

  • List of paradoxes — This is a list of paradoxes, grouped thematically. Note that many of the listed paradoxes have a clear resolution see Quine s Classification of Paradoxes.Logical, non mathematical* Paradox of entailment: Inconsistent premises always make an… …   Wikipedia

  • Список парадоксов — …   Википедия

  • List of probability topics — This is a list of probability topics, by Wikipedia page. It overlaps with the (alphabetical) list of statistical topics. There are also the list of probabilists and list of statisticians.General aspects*Probability *Randomness, Pseudorandomness,… …   Wikipedia

  • Conditional expectation — In probability theory, a conditional expectation (also known as conditional expected value or conditional mean) is the expected value of a real random variable with respect to a conditional probability distribution. The concept of conditional… …   Wikipedia

  • Outline of probability — Probability is the likelihood or chance that something is the case or will happen. Probability theory is used extensively in statistics, mathematics, science and philosophy to draw conclusions about the likelihood of potential events and the… …   Wikipedia

  • Bayes' theorem — In probability theory, Bayes theorem (often called Bayes law after Thomas Bayes) relates the conditional and marginal probabilities of two random events. It is often used to compute posterior probabilities given observations. For example, a… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”