Probability metric

Probability metric

A probability metric is a function defining a distance between random variables or vectors. In particular the probability metric does not satisfy the identity of indiscernibles condition required to be satisfied by the metric of the metric space.

Probability metric of random variables

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

:D(X, Y) = E(|X - Y|).,

If the joint probability distribution is absolutely continuous, this is the same as

:int_{-infty}^infty int_{-infty}^infty |x-y|F(x, y) , dx, dy,

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) = int_{-infty}^infty int_{-infty}^infty |x-y|f(x)g(y) , dx, dy

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

One may easily show that such probability metrics do not satisfy the identity of indiscernibles condition of the metric or satisfies it 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_{deltadelta}(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 mu_x, mu_y of the variables "X" and "Y" and obviously:

:D_{deltadelta}(X, X) = int_{-infty}^infty int_{-infty}^infty |x-x'|delta(x-mu_x)delta(x'-mu_x) , dx, dx' = |mu_x-mu_x| = 0.

For all other cases:

:Dleft(X, X ight) > 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: two continuous random variables with normal distributions (NN)

If both probability distribution functions of random variables "X" and "Y" are normal distributions (N) having the same standard deviation σ, and moreover "X" and "Y" are independent, then evaluating "D"("X", "Y") yields

:D_{NN}(X, Y) = mu_{xy} + frac{2sigma}{sqrtpi}operatorname{exp}left(-frac{mu_{xy}^2}{4sigma^2} ight)-mu_{xy} operatorname{erfc} left(frac{mu_{xy{2sigma} ight)

where:mu_{xy} = left|mu_x-mu_y ight|,

erfc("x") is the complementary error function and subscripts NN indicate the type of the metric.

In this case "zero value" of the probability metric D_{NN}(X, Y) amounts:

:lim_{mu_{xy} o 0} D_{NN}(X, Y) = D_{NN}(X, X) = frac{2sigma}{sqrtpi}.

Example: two continuous random variables with uniform distributions (RR)

In case both random variables "X" and "Y" are characterized by uniform distributions ("R") of the same standard deviation σ, integrating "D"("X", "Y") yields:

:D_{RR}(X, Y) = egin{cases} frac{24sqrt{3}sigma^3-mu_{xy}^3+6sqrt{3}sigmamu_{xy}^2}{36sigma^2}, & mu_{xy}<2sqrt{3}sigma, \\ mu_{xy}, & mu_{xy} ge 2sqrt{3}sigma. end{cases}

The minimal value of this kind of probability metric amounts:

:D_{RR}(X, X) = frac{2sigma}{sqrt{3.

Probability metric of discrete random variables

In case random variables "X" and "Y" are characterized by discrete probability distribution the probability metric "D" may be defined as: :D(X, Y) = sum_{i} sum_{j} |x_i-y_j|P(X=x_i)P(Y=y_j),.

For example for two discrete Poisson-distributed random variables "X" and "Y" the equation above transforms into:

:D_{PP}(X, Y) = sum_{x=0}^nsum_{y=0}^n |x-y|frac sin{left(frac{m pi x}{L} ight)}, ,

:psi_n(y) = sqrt{frac{2}{L sin{left(frac{n pi y}{L} ight)}, ,

may be defined in terms of probability metric of independent random variables as:

:egin{align}&{} D(X, Y) = intlimits_{0}^L intlimits_{0}^L |x-y||psi_m(x)|^2|psi_n(y)|^2 , dx, dy \\&{} = Lleft(frac{1}{3}(m+n)^2 - frac{m^4 + n^4 + 2m^3n + 2mn^3 + 2m^2n^2}{2m^2n^2pi^2} ight).end{align}

The distance between particles "X" and "Y" is obviously minimum for "m" = 1 i "n" = 1, that is for the minimum energy levels of these particles and amounts:

:min(D(X, Y)) = Lleft(frac{4}{3}-frac{4}{pi^2} ight) approx 0.93L ,.

According to the probability metric properties the minimum distance is nonzero. In fact it is close to the length "L" of the potential well. For other energy levels it is even greater than the length of the well.

External references

* [ A new concept of probability metric and its applications in approximation of scattered data sets]

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