Can anyone help me prove under what conditions on the distance function [tex]d(x_1,x_2)[/tex] the following inequality holds for any two probability distributions (represented by probability densities) [tex]p(x)[/tex] and [tex]q(x)[/tex] :(adsbygoogle = window.adsbygoogle || []).push({});

[tex]2\int{\int{d^2(x_1,x_2)p(x_1)q(x_2)dx_1dx_2}

\geq

\int{\int{d^2(x_1,x_2)p(x_1)p(x_2)dx_1dx_2} +

\int{\int{d^2(x_1,x_2)q(x_1)q(x_2)dx_1dx_2}

[/tex]

where [tex]d^2(x_1,x_2)[/tex] is the squared distance between [tex]x_1[/tex] and [tex]x_2[/tex] in some metric space [tex]\Theta[/tex]. All integrals are over [tex]\Theta[/tex].

One can easily verify by insertion that the inequality holds for a Euclidian metric where [tex]d^2(x_1,x_2)=(x_1-x_2)^2[/tex], with equality if and only if the expectation of [tex]p(x)[/tex] and [tex]q(x)[/tex] are the same.

It must surely hold for some more general class of metrics (described by [tex]d^2(x_1,x_2)[/tex]) - possibly all metrics - but I've so far failed to demonstrate it. Does anyone have an idea of how to prove it in some more general case?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Probability theoretic inequality

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**