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I know that the maximum possible Shannon Entropy for an alphabet $X$ is $\log|X|$, where Shannon Entropy is:

$$H(X) = - \sum_{x \in X} \; p(x) \log p(x)$$

but how is this upper limit computed?

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No, its $\log |X|$ if $p(x)>0$ for all $x\in X$; otherwise its $|\{x:p(x)>0\}|$. Write $H(X) = \sum_{x \in X} \; p(x) \log(1/ p(x))$, and then use Jensen's inequality. –  Ashok Jul 9 '12 at 11:06
    
Good point, stupid fingers - will update the question to fix the $log|X|$ issue. I was trying to differentiate and set to zero, I think I see how this approach works, thank you –  Brabster Jul 9 '12 at 11:17
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