<p>It's the <span class="math-container">$\alpha^{\mathrm{th}}$</span> moment of the Tribus <a href="https://en.wikipedia.org/wiki/Information_content#cite_note-Tribus1961-6" rel="nofollow noreferrer">surprisal</a>.</p>
<p>This generalizes the statement that <em>entropy = expected surprisal</em>.</p>
<p>Or in <a href="http://julio.staff.ipb.ac.id/files/2015/02/Ross_8th_ed_English.pdf" rel="nofollow noreferrer">Ross's textbook</a>, &quot;expected surprise&quot;.</p>

CC BY-SA 4.0
2020-09-23T19:49:40.680Z
So... "moment of surprise"?
Bjørn Kjos-Hanssen
CC BY-SA 4.0
2020-09-23T22:33:03.483Z
I searched Ross's book for "expected surprise" and nothing came up -- what page is it on?
Aryeh
CC BY-SA 4.0
2020-09-23T22:38:21.593Z
Any chance you have Trubus's book in pdf?
Aryeh
CC BY-SA 4.0
2020-09-24T02:39:14.347Z
You can search for "surprise"
Bjørn Kjos-Hanssen
CC BY-SA 4.0
2020-09-23T19:42:00.960Z
2020-09-23T19:42:00.960Z
Bjørn Kjos-Hanssen
<p>So we ended up calling this quantity <em>the <span class="math-container">$\alpha$</span>th moment of information</em> and proving some inequalities about it:
<a href="https://arxiv.org/abs/2004.12680" rel="nofollow noreferrer">https://arxiv.org/abs/2004.12680</a>
(paper to appear in the NIPS 2021 conference).</p>

CC BY-SA 4.0
2021-09-30T15:55:23.233Z
2021-09-30T15:55:23.233Z
Aryeh
<p>For <span class="math-container">$p\in[0,1]^{\mathbb{N}}$</span> and <span class="math-container">$\alpha\ge1$</span>, define
<span class="math-container">$$ H_\alpha(p) = \sum_{i\in\mathbb{N}}p_i|\log(p_i)|^\alpha.
$$</span>
When <span class="math-container">$\sum_i p_i=1$</span> and <span class="math-container">$\alpha=1$</span>, <span class="math-container">$H_1(p)$</span> is just the Shannon entropy of the distribution <span class="math-container">$p$</span>. Has anyone encountered the object <span class="math-container">$H_\alpha$</span> anywhere in the literature? A reference would be much appreciated. I'm thinking of calling <span class="math-container">$H_\alpha$</span> <em>hyperentropy</em> for <span class="math-container">$\alpha&gt;1$</span>; is <em>that</em> term already taken by chance?</p>

CC BY-SA 4.0
2020-09-21T23:19:30.087Z
Not an answer, but linking https://math.stackexchange.com/questions/3781318/name-of-a-quantity-related-to-the-shannon-entropy
Clement C.
CC BY-SA 4.0
2020-09-21T23:20:36.947Z
Thanks!! I'm curious about the *other* guy's motivation...
Aryeh
<reference-request><shannon-entropy>
Entropy-like quantity
Aryeh
# Entropy-like quantity

(Asked by: username_0 on 2020-09-21)

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 $\alpha>1$; is *that* term already taken by chance?

**Tags:** reference-request, shannon-entropy
> **Comments:**
> **username_1**: Not an answer, but linking https://math.stackexchange.com/questions/3781318/name-of-a-quantity-related-to-the-shannon-entropy  
> **username_0**: Thanks!! I'm curious about the *other* guy's motivation...  

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## 2 answers

### Answer by username_2 on 2020-09-23. Score: 2 | Accepted answer

It's the $\alpha^{\mathrm{th}}$ moment of the Tribus [surprisal](https://en.wikipedia.org/wiki/Information_content#cite_note-Tribus1961-6).

This generalizes the statement that *entropy = expected surprisal*.

Or in [Ross's textbook](http://julio.staff.ipb.ac.id/files/2015/02/Ross_8th_ed_English.pdf), "expected surprise".

> **Comments:**
> **username_2**: So... "moment of surprise"?  
> **username_0**: I searched Ross's book for "expected surprise" and nothing came up -- what page is it on?  
> **username_0**: Any chance you have Trubus's book in pdf?  
> **username_2**: You can search for "surprise"  

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### Answer by username_0 on 2021-09-30. Score: 1

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).


---

