Puneet Varma (Editor)

Probabilistic metric space

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A probabilistic metric space is a generalization of metric spaces where the distance is no longer defined on positive real numbers, but on distribution functions.

Contents

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 sup(F(x)) = 1 for x∈R.

The ordered pair (S,F) is said to be a probabilistic metric space if S is a nonempty set and F: S×S → D+ (F(p, q) is denoted by Fp,q for every (p, q) ∈ S × S) satisfies the following conditions:

  • Fu,v(x) = 1 for every x > 0 ⇔ u = v (u, v ∈ S).
  • Fu,v = Fv,u for every u, v ∈ S.
  • Fu,v(x) = 1 and Fv,w(y) = 1 ⇒ Fu,w(x + y) = 1 f or u, v, w ∈ S and x, y ∈ R+.

    • Probability metric of random variables

    A probability metric D between two random variables X and Y may be defined e.g. as:

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

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

    D ( X , Y ) = ∫ − ∞ ∞ ∫ − ∞ ∞ | x − y | f ( x ) g ( y ) d x d y

    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 is only if, and only if, both of its arguments X, Y are certain events described by Dirac delta density probability distribution functions. In this case:

    D ( X , Y ) = ∫ − ∞ ∞ ∫ − ∞ ∞ | x − y | δ ( x − μ x ) δ ( y − μ y ) d x d y = | μ x − μ y |

    the probability metric simply transforms into the metric between expected values μ x , μ 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

    Example

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

    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 )

    where:

    μ x y = | μ x − μ y | ,

    and erfc ⁡ ( x ) is the complementary error function.

    In this case:

    lim μ x y → 0 D N N ( X , Y ) = D N N ( X , X ) = 2 σ π

    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 ( X , Y ) = ∫ Ω ∫ Ω d ( x , y ) F ( x , y ) d Ω x d Ω 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 ( X , Y ) = ∫ Ω ∫ Ω ∑ i | x i − y i | 2 F ( x ) G ( y ) d Ω x d Ω y

    References

    Probabilistic metric space Wikipedia
    (Text) CC BY-SA