Concavity of Entropy: Is it True?

[emma83]

Homework Statement

Shannon entropy is a concave function defined as follows:
H(X)=-\sum_{x}p(x)\log p(x)

Conditional Shannon entropy is defined as follows:
H(X|Y)=\sum_{y} p(y) H(X|Y=y)=-\sum_{y} p(y)\sum_{x}p(x|Y=y)\log p(x|Y=y)

Can we deduce that:
\sum_{y} p(y)H(X|Y=y)\geq H(X|Y=y)

Homework Equations

The Attempt at a Solution

I would say yes because of the concavity but I am confused with the 2 random variables.

 

[jmb]

I'm assuming the RHS of your final expression is also meant to be summed over y, otherwise it doesn't make much sense...

The way to go about this is to think about the nature of p(y). What constraints do you know about the values that p(y) can take?

PS Do you really mean \geq in the last line, or do you mean \leq?