TITLE: Geometric interpretations of the exponential of entropy
QUESTION [14 upvotes]: Question:
Might there be a natural geometric interpretation of the exponential of entropy in Classical and Quantum Information theory? This question occurred to me recently via a geometric inequality concerning the exponential of the Shannon entropy.
Original motivation:
The weighted AM-GM inequality states that if $\{a_i\}_{i=1}^n,\{\lambda_i\}_{i=1}^n \in \mathbb{R}_+^n$ and $\sum_{i=1}^n \lambda_i = 1$, then:
\begin{equation}
\prod_{i=1}^n a_i^{\lambda_i} \leq \sum_{i=1}^n \lambda_i \cdot a_i \tag{1}
\end{equation}
As an application, we find that if $H(\vec{p})$ denotes the Shannon entropy of a discrete probability distribution $\vec{p} = \{p_i\}_{i=1}^n$ and $r_p^2 = \lVert \vec{p} \rVert^2 $ is the $l_2$ norm of $\vec{p}$ then:
\begin{equation}
e^{H(\vec{p})} \geq \frac{1}{r_p^2} \tag{2}
\end{equation}
This result follows from the observation that if $a_i = p_i$ and $\lambda_i = p_i$,
\begin{equation}
e^{-H(\vec{p})} = e^{\sum_i p_i \ln p_i} = \prod_{i=1}^n p_i^{p_i} \tag{3}
\end{equation}
\begin{equation}
\sum_{i=1}^n p_i^2 = \lVert \vec{p} \rVert^2 \tag{4}
\end{equation}
and using (1), we may deduce (2) where equality is obtained when the Shannon entropy is maximised by the uniform distribution i.e. $\forall i, p_i = \frac{1}{n}$.
A remark on appropriate geometric embeddings:
If we consider that the Shannon entropy measures the quantity of hidden information in a stochastic system at the state $\vec{p} \in [0,1]^n$, we may define the level sets $\mathcal{L}_q$ in terms of the typical probability $q \in (0,1)$:
\begin{equation}
\mathcal{L}_q = \{\vec{p} \in [0,1]^n: e^{H(\vec{p})} = e^{- \ln q} \} \tag{5}
\end{equation}
which allows us to define an equivalence relation over states $\vec{p} \in [0,1]^n$. Such a model is appropriate for events which may have $n$ distinct outcomes.
Now, we'll note that $e^{H(\vec{p})}$ has a natural interpretation as a measure of hidden information while $e^{-H(\vec{p})}$ may be interpreted as the typical probability of the state $\vec{p}$. Given (5), a natural relation between these measures may be found using the Hyperbolic identities:
\begin{equation}
\cosh^2(-\ln q) - \sinh^2(-\ln q) = 1 \tag{6}
\end{equation}
\begin{equation}
\cosh(-\ln q) - \sinh(-\ln q) = q \tag{7}
\end{equation}
where $2 \cdot \cosh(-\ln q)$ is the sum of these two measures and $2 \cdot \sinh(-\ln q)$ may be understood as their difference. This suggests that the level sets $\mathcal{L}_q$ have a natural Hyperbolic embedding in terms of Hyperbolic functions.
References:

Olivier Rioul. This is IT: A Primer on Shannon’s Entropy and Information. Séminaire Poincaré. 2018.

David J.C. MacKay. Information Theory, Inference and Learning Algorithms. Cambridge University Press 2003.

John C. Baez, Tobias Fritz, Tom Leinster. A Characterization of Entropy in Terms of Information Loss. Arxiv. 2011.

REPLY [2 votes]: With apologies also for promoting my own work, a colleague (Tim Hosgood) and I have recently published an article in Entropy, where we give a geometric interpretation for the exponentiated entropy of empirical distributions, namely as the length $L$ of a certain rectangle. What makes this rectangle interesting is that its width $W$ and area $A$ (from which you can calculate its length as $L=\frac{A}{W}$ and hence entropy as $H=\log(L)$) are additively and multiplicatively homomorphic in the original distribution.
It's well known that the entropy of a product of distributions is the sum of their entropies, so the exponentiated entropy (sometimes called the perplexity) of a product is the product of the exponentiated entropies. Thus one says that exponentiated entropy is multiplicatively homomorphic in the distribution. What we're saying in this short note is that if you use the whole rectangle (its area and width) instead of just its length, you can get not only a multiplicative but also an additive homomorphism; that is you get a rig homomorphism from empirical distributions to rectangles.