Probabilistic metric space

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In mathematics, probabilistic metric spaces are a generalization of metric spaces where the distance no longer takes values in the non-negative real numbers R_≥ 0, but in distribution functions.^[1]

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 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.^[2]

Probabilistic metric spaces are initially introduced by Menger, which were termed statistical metrics.^[3] Shortly after, Wald criticized the generalized triangle inequality and proposed an alternative one.^[4] However, both authors had come to the conclusion that in some respects the Wald inequality was too stringent a requirement to impose on all probability metric spaces, which is partly included in the work of Schweizer and Sklar.^[5] Later, the probabilistic metric spaces found to be very suitable to be used with fuzzy sets^[6] and further called fuzzy metric spaces^[7]

A probability metric D between two random variables X and Y may be defined, for example, as $${\displaystyle D(X,Y)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }|x-y|F(x,y)\,dx\,dy}$$ 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 $${\displaystyle D(X,Y)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }|x-y|f(x)g(y)\,dx\,dy}$$ 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: $${\displaystyle D(X,Y)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }|x-y|\delta (x-\mu _{x})\delta (y-\mu _{y})\,dx\,dy=|\mu _{x}-\mu _{y}|}$$ the probability metric simply transforms into the metric between expected values ${\displaystyle \mu _{x}}$, ${\displaystyle \mu _{y}}$ 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: $${\displaystyle D\left(X,X\right)>0.}$$

For example if both probability distribution functions of random variables X and Y are normal distributions (N) having the same standard deviation ${\displaystyle \sigma }$, integrating ${\displaystyle D\left(X,Y\right)}$ yields: $${\displaystyle D_{NN}(X,Y)=\mu _{xy}+{\frac {2\sigma }{\sqrt {\pi }}}\exp \left(-{\frac {\mu _{xy}^{2}}{4\sigma ^{2}}}\right)-\mu _{xy}\operatorname {erfc} \left({\frac {\mu _{xy}}{2\sigma }}\right)}$$ where $${\displaystyle \mu _{xy}=\left|\mu _{x}-\mu _{y}\right|,}$$ and ${\displaystyle \operatorname {erfc} (x)}$ is the complementary error function.

In this case: $${\displaystyle \lim _{\mu _{xy}\to 0}D_{NN}(X,Y)=D_{NN}(X,X)={\frac {2\sigma }{\sqrt {\pi }}}.}$$

The probability metric of random variables may be extended into metric D(X, Y) of random vectors X, Y by substituting ${\displaystyle |x-y|}$ with any metric operator d(x, y): $${\displaystyle D(\mathbf {X} ,\mathbf {Y} )=\int _{\Omega }\int _{\Omega }d(\mathbf {x} ,\mathbf {y} )F(\mathbf {x} ,\mathbf {y} )\,d\Omega _{x}d\Omega _{y}}$$ 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: $${\displaystyle D(\mathbf {X} ,\mathbf {Y} )=\int _{\Omega }\int _{\Omega }{\sqrt {\sum _{i}|x_{i}-y_{i}|^{2}}}F(\mathbf {x} )G(\mathbf {y} )\,d\Omega _{x}d\Omega _{y}.}$$