probabilistic metric space

Probabilistic metric spaces are initially introduced by Menger, which were termed statistical metrics.{{Citation | vauthors=((Menger, K.)) | title=Selecta Mathematica | year=2003 | chapter=Statistical Metrics | pages=433–435 | publisher=Springer Vienna | doi=10.1007/978-3-7091-6045-9_35 | isbn=978-3-7091-7294-0 | chapter-url=http://dx.doi.org/10.1007/978-3-7091-6045-9_35}} Shortly after, Wald criticized the generalized triangle inequality and proposed an alternative one.{{Citation | vauthors=((Wald, A.)) | year=1943 | title=On a Statistical Generalization of Metric Spaces | journal=Proceedings of the National Academy of Sciences | volume=29 | issue=6 | pages=196–197 | doi=10.1073/pnas.29.6.196 | doi-access=free | pmid=16578072 | pmc=1078584 | bibcode=1943PNAS...29..196W }} 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.{{Citation | vauthors=((Schweizer, B. and Sklar, A)) | chapter=Statistical Metrics | date=2003 | pages=433–435 | title=Selecta Mathematica | publisher=Springer Vienna | doi=10.1007/978-3-7091-6045-9_35 | isbn=978-3-7091-7294-0 | url=http://dx.doi.org/10.1007/978-3-7091-6045-9_35}} Later, the probabilistic metric spaces found to be very suitable to be used with fuzzy sets{{cite book | vauthors=((Bede, B.)) | date= 2013 | title=Mathematics of Fuzzy Sets and Fuzzy Logic | series= Studies in Fuzziness and Soft Computing | volume= 295 | publisher=Springer Berlin Heidelberg | url=http://dx.doi.org/10.1007/978-3-642-35221-8 | doi=10.1007/978-3-642-35221-8| isbn= 978-3-642-35220-1 }} and further called fuzzy metric spaces{{Cite journal |last1=Kramosil |first1=Ivan |last2=Michálek |first2=Jiří |date=1975 |title=Fuzzy metrics and statistical metric spaces |url=https://www.kybernetika.cz/content/1975/5/336/paper.pdf |journal=Kybernetika |volume=11 |issue=5 |pages=336–344}}

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

$$D(X,\; Y)\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash int\_\{-\backslash infty\}^\backslash infty\; |x-y|\; F(x,\; y)\; \backslash ,\; dx\; \backslash ,\; dy$$

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

$$D(X,\; Y)\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash int\_\{-\backslash infty\}^\backslash infty\; |x-y|\; f(x)\; g(y)\; \backslash ,\; dx\; \backslash ,\; dy$$

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:

$$D(X,\; Y)\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash int\_\{-\backslash infty\}^\backslash infty\; |x-y|\; \backslash delta(x-\backslash mu\_x)\; \backslash delta(y-\backslash mu\_y)\; \backslash ,\; dx\; \backslash ,\; dy\; =\; |\backslash mu\_x\; -\; \backslash mu\_y|$$

the probability metric simply transforms into the metric between expected values $\backslash mu\_x$, $\backslash mu\_y$ 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:

$$D\backslash left(X,\; X\backslash right)\; >\; 0.$$

[[Image:Probability_metric_DNN.png|thumb|right|325px|

Probability metric between two random variables X and Y, both having normal distributions and the same standard deviation $\backslash sigma\; =\; 0,\; \backslash sigma\; =\; 0.2,\; \backslash sigma\; =\; 0.4,\; \backslash sigma\; =\; 0.6,\; \backslash sigma\; =\; 0.8,\; \backslash sigma\; =\; 1$ (beginning with the bottom curve).

$m\_\{xy\}\; =\; |\backslash mu\_x\; -\; \backslash mu\_y|$ denotes a distance between means of X and Y.]]

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

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

where

$$\backslash mu\_\{xy\}\; =\; \backslash left|\backslash mu\_x\; -\; \backslash mu\_y\backslash right|,$$

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

In this case:

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

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):

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

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:

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

{{Reflist}}

Category:Probability distributions

Category:Metric geometry