mutual information. concave/convex

PetitPrince

hi everybody,

while looking on the mutual information of two variables, one find that it is concave of p(x) given p(x|y) and convex of p(x|y) given p(x).

the first statement is okey, but when it comes to proving the second, i get stuck, even when i find proofs already done i didn't get how they can conclude the convexity of I(x,y) as a function of p(x|y) from the convexity of the relative entropy D(p||q).

here is a piece of the proof i didnt understand
http://ocw.usu.edu/Electrical_and_Co...y/lecture3.pdf

if you have any idea, i'd very much appreciate it.

thank you in advance.