# Is conditional entropy a convex function?

A conditional entropy can be expressed in the following way, $H_{V_t}(V_s) = -\sum_{s,t}p(s,t)\log{p_t(s)} = -\sum_{s,t}p(s,t)\log{\frac{p(s,t)}{\sum_{s'}{p(s',t)}}}$

$s$ and $t$ are defined on finite domain, say $s,t\in{\{00,01,11,10\}}$. Each value of $p(s,t)$ is an independent variable. Is conditional entropy a convex function under this circumstances?

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As it stands, I can see three disconcerting things: 1) I think there is a minus missing in the formula of conditional entropy. 2) If $p_s(t)$ stands for the conditional probability, then the denominator in the last expression should be summed over the other variable (here $t$) 3) A function is convex in some variable. The variable is not clear here. –  Ashok Feb 9 '12 at 6:36
variable is $p(s,t)$, not $s$ and $t$. The question is also improved accordingly. –  Richard Feb 13 '12 at 21:37