Probabilistic metric space

In mathematics, probabilistic metric spaces is a generalizations of metric spaces where the distance no longer takes values in the non-negative real numbersc R_≥ 0, but in distribution functions. Let D+ be the set of all probability distribution functions F such that F(0) = 0 (F is a nondecreasing, left continuous mapping from R into [0, 1] such that max(F) = 1).

Then given a non-empty set S and a function F: S × S → D+ where we denote F(p, q) by F_p,q for every (p, q) ∈ S × S, the ordered pair (S, F) is said to be a probabilistic metric space if:

• For all u and v in S, u = v if and only if F_u,v(x) = 1 for all x > 0.
• For all u and v in S, F_u,v = F_v,u.
• For all u, v and w in S, F_u,v(x) = 1 and F_v,w(y) = 1 ⇒ F_u,w(x + y) = 1 for x, y > 0.

Probability metric of random variables

A probability metric D between two random variables X and Y may be defined, for example, as

$D(X,Y)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}|x-y|F(x,y)dxdy$

where F(x, y) denotes the joint probability density function of the random variables X and Y. If X and Y are independent from each other then the equation above transforms into

$D(X,Y)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}|x-y|f(x)g(y)dxdy$

where f(x) and g(y) are probability density functions of X and Y respectively.

One may easily show that such probability metrics do not satisfy the first metric axiom or satisfies it if, and only if, both of arguments X and Y are certain events described by Dirac delta density probability distribution functions. In this case:

$D(X,Y)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}|x-y|\delta (x-{\mu }_x)\delta (y-{\mu }_y)dxdy=|{\mu }_x-{\mu }_y|$

the probability metric simply transforms into the metric between expected values ${\mu }_x$, ${\mu }_y$ of the variables X and Y.

For all other random variables X, Y the probability metric does not satisfy the identity of indiscernibles condition required to be satisfied by the metric of the metric space, that is:

$D(X,X)>0.$

Probability metric between two random variables X and Y, both having normal distributions and the same standard deviation $\sigma =0,\sigma =0.2,\sigma =0.4,\sigma =0.6,\sigma =0.8,\sigma =1$ (beginning with the bottom curve). $m_{xy}=|{\mu }_x-{\mu }_y|$ denotes a distance between means of X and Y.

Example

For example if both probability distribution functions of random variables X and Y are normal distributions (N) having the same standard deviation $\sigma$, integrating $D(X,Y)$ yields:

$D_{NN}(X,Y)={\mu }_{xy}+\frac{2\sigma }{\sqrt{\pi }}\exp (-\frac{{\mu }_{xy}^2}{4{\sigma }^2})-{\mu }_{xy}\mathrm{erfc}\left(\frac{{\mu }_{xy}}{2\sigma }\right)$

where

${\mu }_{xy}=|{\mu }_x-{\mu }_y|$,

and $\mathrm{erfc}(x)$ is the complementary error function.

In this case:

$\underset{{\mu }_{xy}\to 0}{\lim }{D}_{NN}(X,Y)=D_{NN}(X,X)=\frac{2\sigma }{\sqrt{\pi }}.$

Probability metric of random vectors

The probability metric of random variables may be extended into metric D(X, Y) of random vectors X, Y by substituting $|x-y|$ with any metric operator d(x, y):

$D(\mathbf{𝐗},\mathbf{𝐘})=\underset{\mathrm{\Omega }}{\int }\underset{\mathrm{\Omega }}{\int }d(\mathbf{𝐱},\mathbf{𝐲})F(\mathbf{𝐱},\mathbf{𝐲})d{\mathrm{\Omega }}_{x}d{\mathrm{\Omega }}_y$

where F(X, Y) is the joint probability density function of random vectors X and Y. For example substituting d(x, y) with Euclidean metric and providing the vectors X and Y are mutually independent would yield to:

$D(\mathbf{𝐗},\mathbf{𝐘})=\underset{\mathrm{\Omega }}{\int }\underset{\mathrm{\Omega }}{\int }\sqrt{\sum _i|x_i-y_i{|}^2}F(\mathbf{𝐱})G(\mathbf{𝐲})d{\mathrm{\Omega }}_{x}d{\mathrm{\Omega }}_y.$