Probabilistic metric space

<h2 id="history">History</h2>
Probabilistic metric spaces are initially introduced by Menger, which were termed statistical metrics.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> Shortly after, Wald criticized the generalized <a href="/facts/Triangle_inequality/KubLjkwr">triangle inequality</a> and proposed an alternative one.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> 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.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> Later, the probabilistic metric spaces found to be very suitable to be used with fuzzy sets<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> and further called fuzzy metric spaces<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

<h2 id="probability-metric-of-random-variables">Probability metric of random variables</h2>
A probability metric D between two <a href="/facts/Random_variable/TwTBXnLT">random variables</a> X and Y may be defined, for example, as

D
        (
        X
        ,
        Y
        )
        =
        
          ∫
          
            −
            ∞
          
          
            ∞
          
        
        
          ∫
          
            −
            ∞
          
          
            ∞
          
        
        
          |
        
        x
        −
        y
        
          |
        
        F
        (
        x
        ,
        y
        )
        
        d
        x
        
        d
        y
      
    
    {\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

D
        (
        X
        ,
        Y
        )
        =
        
          ∫
          
            −
            ∞
          
          
            ∞
          
        
        
          ∫
          
            −
            ∞
          
          
            ∞
          
        
        
          |
        
        x
        −
        y
        
          |
        
        f
        (
        x
        )
        g
        (
        y
        )
        
        d
        x
        
        d
        y
      
    
    {\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 <a href="/facts/Metric_(mathematics)/RkUptSo8">metric</a> axiom or satisfies it <a href="/facts/If_and_only_if/bYSxGJ66">if, and only if</a>, both of arguments X and Y are certain events described by <a href="/facts/Dirac_delta/V8PjRP8T">Dirac delta</a> density <a href="/facts/Probability_density_function/zvfybna4">probability distribution functions</a>. In this case:

D
        (
        X
        ,
        Y
        )
        =
        
          ∫
          
            −
            ∞
          
          
            ∞
          
        
        
          ∫
          
            −
            ∞
          
          
            ∞
          
        
        
          |
        
        x
        −
        y
        
          |
        
        δ
        (
        x
        −
        
          μ
          
            x
          
        
        )
        δ
        (
        y
        −
        
          μ
          
            y
          
        
        )
        
        d
        x
        
        d
        y
        =
        
          |
        
        
          μ
          
            x
          
        
        −
        
          μ
          
            y
          
        
        
          |
        
      
    
    {\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 <a href="/facts/Expected_value/1XV0JKL8">expected values</a> 
 
 
 
 
 μ
 
 x
 
 
 
 
 {\displaystyle \mu _{x}}
 
, 
 
 
 
 
 μ
 
 y
 
 
 
 
 {\displaystyle \mu _{y}}
 
 of the variables X and Y.
For all other <a href="/facts/Random_variable/TwTBXnLT">random variables</a> X, Y the probability metric does not satisfy the <a href="/facts/Identity_of_indiscernibles/uaDTDeq8">identity of indiscernibles</a> condition required to be satisfied by the metric of the metric space, that is:

D
        
          (
          
            X
            ,
            X
          
          )
        
        >
        0.
      
    
    {\displaystyle D\left(X,X\right)>0.}

<h3>Example</h3>
For example if both <a href="/facts/Probability_density_function/zvfybna4">probability distribution functions</a> of random variables X and Y are <a href="/facts/Normal_distribution/UapjjPyQ">normal distributions</a> (N) having the same <a href="/facts/Standard_deviation/qSug9BRl">standard deviation</a> 
 
 
 
 σ
 
 
 {\displaystyle \sigma }
 
, integrating 
 
 
 
 D
 
 (
 
 X
 ,
 Y
 
 )
 
 
 
 {\displaystyle D\left(X,Y\right)}
 
 yields:

D
          
            N
            N
          
        
        (
        X
        ,
        Y
        )
        =
        
          μ
          
            x
            y
          
        
        +
        
          
            
              2
              σ
            
            
              π
            
          
        
        exp
        ⁡
        
          (
          
            −
            
              
                
                  μ
                  
                    x
                    y
                  
                  
                    2
                  
                
                
                  4
                  
                    σ
                    
                      2
                    
                  
                
              
            
          
          )
        
        −
        
          μ
          
            x
            y
          
        
        erfc
        ⁡
        
          (
          
            
              
                μ
                
                  x
                  y
                
              
              
                2
                σ
              
            
          
          )
        
      
    
    {\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

μ
          
            x
            y
          
        
        =
        
          |
          
            
              μ
              
                x
              
            
            −
            
              μ
              
                y
              
            
          
          |
        
        ,
      
    
    {\displaystyle \mu _{xy}=\left|\mu _{x}-\mu _{y}\right|,}

and 
 
 
 
 erfc
 ⁡
 (
 x
 )
 
 
 {\displaystyle \operatorname {erfc} (x)}
 
 is the complementary <a href="/facts/Error_function/Ca2H7pGr">error function</a>.
In this case:

lim
          
            
              μ
              
                x
                y
              
            
            →
            0
          
        
        
          D
          
            N
            N
          
        
        (
        X
        ,
        Y
        )
        =
        
          D
          
            N
            N
          
        
        (
        X
        ,
        X
        )
        =
        
          
            
              2
              σ
            
            
              π
            
          
        
        .
      
    
    {\displaystyle \lim _{\mu _{xy}\to 0}D_{NN}(X,Y)=D_{NN}(X,X)={\frac {2\sigma }{\sqrt {\pi }}}.}

<h2 id="probability-metric-of-random-vectors">Probability metric of random vectors</h2>
The probability metric of random variables may be extended into metric D(X, Y) of <a href="/facts/Random_vector/qMfooyVf">random vectors</a> X, Y by substituting 
 
 
 
 
 |
 
 x
 −
 y
 
 |
 
 
 
 {\displaystyle |x-y|}
 
 with any metric operator d(x, y):

D
        (
        
          X
        
        ,
        
          Y
        
        )
        =
        
          ∫
          
            Ω
          
        
        
          ∫
          
            Ω
          
        
        d
        (
        
          x
        
        ,
        
          y
        
        )
        F
        (
        
          x
        
        ,
        
          y
        
        )
        
        d
        
          Ω
          
            x
          
        
        d
        
          Ω
          
            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 <a href="/facts/Euclidean_metric/9qDoQKQe">Euclidean metric</a> and providing the vectors X and Y are mutually independent would yield to:

D
        (
        
          X
        
        ,
        
          Y
        
        )
        =
        
          ∫
          
            Ω
          
        
        
          ∫
          
            Ω
          
        
        
          
            
              ∑
              
                i
              
            
            
              |
            
            
              x
              
                i
              
            
            −
            
              y
              
                i
              
            
            
              
                |
              
              
                2
              
            
          
        
        F
        (
        
          x
        
        )
        G
        (
        
          y
        
        )
        
        d
        
          Ω
          
            x
          
        
        d
        
          Ω
          
            y
          
        
        .
      
    
    {\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}.}

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Sherwood, H. (1971). "Complete probabilistic metric spaces". Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete. 20 (2): 117–128. doi:10.1007/bf00536289. ISSN 0044-3719. <a href="https://doi.org/10.1007%2Fbf00536289" target="_blank">https://doi.org/10.1007%2Fbf00536289</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Schweizer, Berthold; Sklar, Abe (1983). Probabilistic metric spaces. North-Holland series in probability and applied mathematics. New York: North-Holland. ISBN 978-0-444-00666-0. <a href="978-0-444-00666-0" target="_blank">978-0-444-00666-0</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Menger, K. (2003), "Statistical Metrics", Selecta Mathematica, Springer Vienna, pp. 433–435, doi:10.1007/978-3-7091-6045-9_35, ISBN 978-3-7091-7294-0 <a href="978-3-7091-7294-0" target="_blank">978-3-7091-7294-0</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Wald, A. (1943), "On a Statistical Generalization of Metric Spaces", Proceedings of the National Academy of Sciences, 29 (6): 196–197, Bibcode:1943PNAS...29..196W, doi:10.1073/pnas.29.6.196, PMC 1078584, PMID 16578072 <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Schweizer, B. and Sklar, A (2003), "Statistical Metrics", Selecta Mathematica, Springer Vienna, pp. 433–435, doi:10.1007/978-3-7091-6045-9_35, ISBN 978-3-7091-7294-0 <a href="978-3-7091-7294-0" target="_blank">978-3-7091-7294-0</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Bede, B. (2013). Mathematics of Fuzzy Sets and Fuzzy Logic. Studies in Fuzziness and Soft Computing. Vol. 295. Springer Berlin Heidelberg. doi:10.1007/978-3-642-35221-8. ISBN 978-3-642-35220-1. <a href="978-3-642-35220-1" target="_blank">978-3-642-35220-1</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Kramosil, Ivan; Michálek, Jiří (1975). "Fuzzy metrics and statistical metric spaces" (PDF). Kybernetika. 11 (5): 336–344. <a href="https://www.kybernetika.cz/content/1975/5/336/paper.pdf" target="_blank">https://www.kybernetika.cz/content/1975/5/336/paper.pdf</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
</ol>

Probabilistic metric space open-in-new

Probabilistic metric space