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 - The Fusion of Science and Community**

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

Loading...

Similar Threads - Probability theoretic inequality | Date |
---|---|

I Probability of getting 3 out of 4 numbers correct | Today at 2:40 AM |

B Probability and Death Sentences | Thursday at 10:48 PM |

B Probability of loto hitting a specific place | Mar 10, 2018 |

Conflict between experimental and theoretical probability results | Mar 23, 2013 |

Empirical and theoretical probability | Aug 14, 2011 |

**Physics Forums - The Fusion of Science and Community**