The relevance of Shannon entropy is to repeated sampling: Given $n$ independent samples from a source with binary Shannon Entropy $k$, you can extract $nk(1+o(1)$ i.i.d. uniform bits as $n$ tends to infinity with probability tending to 1. This follows e.g. from the Keane-Smorodinsky [1] finitary isomorphism theorem. See also [2]-[5] below. [1] M. Keane and M....


This is called "output entropy". Suppose you have a communications channel that takes a string and then outputs a random string within a relative $\delta$ radius of it. If you use the input distribution $X$ for communicating over this channel, the entropy of the output distribution would be the quantity you are after.


So we ended up calling this quantity the $\alpha$th moment of information and proving some inequalities about it: https://arxiv.org/abs/2004.12680 (paper to appear in the NIPS 2021 conference).

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