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If I want to use entropy as an approximation for the expected length of Huffman coding, how good is the approximation? I know the following identity:

$H(X)\leq L(X)< H(X)+1\ $

where expected length is:

$L(X)= \sum_i p_il_i$

and entropy is:

$H(X)= \sum_i p_ilog_2(1/p_i)$

But it's not tight enough, or in other words, helpful enough in some cases. Is there a way to approximate the residual using $p_i$, like what we do in Taylor series?

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    $\begingroup$ "But it's not tight enough". Well, in some sense it is. Take for example a Bernoulli variable with $p \approx 0$... $\endgroup$ – leonbloy Jan 6 at 14:09

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