Entropy-like quantity For $p\in[0,1]^{\mathbb{N}}$ and $\alpha\ge1$, define
$$ H_\alpha(p) = \sum_{i\in\mathbb{N}}p_i|\log(p_i)|^\alpha.
$$
When $\sum_i p_i=1$ and $\alpha=1$, $H_1(p)$ is just the Shannon entropy of the distribution $p$. Has anyone encountered the object $H_\alpha$ anywhere in the literature? A reference would be much appreciated. I'm thinking of calling $H_\alpha$ hyperentropy for $\alpha&gt;1$; is that term already taken by chance?
It's the $\alpha^{\mathrm{th}}$ moment of the Tribus surprisal.
This generalizes the statement that entropy = expected surprisal.
Or in Ross's textbook, &quot;expected surprise&quot;.