winterfors
Jul21-08, 07:00 PM
Can anyone help me prove under what conditions on the distance function d(x_1,x_2) the following inequality holds for any two probability distributions (represented by probability densities) p(x) and q(x) :
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}
where d^2(x_1,x_2) is the squared distance between x_1 and x_2 in some metric space \Theta. All integrals are over \Theta.
One can easily verify by insertion that the inequality holds for a Euclidian metric where d^2(x_1,x_2)=(x_1-x_2)^2, with equality if and only if the expectation of p(x) and q(x) are the same.
It must surely hold for some more general class of metrics (described by d^2(x_1,x_2)) - 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?
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}
where d^2(x_1,x_2) is the squared distance between x_1 and x_2 in some metric space \Theta. All integrals are over \Theta.
One can easily verify by insertion that the inequality holds for a Euclidian metric where d^2(x_1,x_2)=(x_1-x_2)^2, with equality if and only if the expectation of p(x) and q(x) are the same.
It must surely hold for some more general class of metrics (described by d^2(x_1,x_2)) - 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?