Generalized entropies
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\newcommand{\gtttlog}{\slog\gttt}$
Entropies are dimensionless notions of how spread out a random variable is (as opposed to dispersions, which care about distance in space). Concretely, for any convex function $f$ such that $f(0) = f(1)= 0$ we can define the $f$-entropy of a distribution $P \in \tri(X)$ as
\[
\al{
H^f(P)
&\ce -\sum_{x \in X} f(P(x)) = -\E_{\Bx \sim P}\b{\f{f(P(\Bx))}{P(\Bx)}}.
}
\]
Range
By convexity, $f(p) \le 0$ for any $p \in [0,1]$, so $H^f(P)$ is nonnegativew. On the other hand, if we let $n \ce |X|$, then by convexity we have
\[
\al{
\f{H^f(P)}{n}&= -\E_{\Bx \in X}\b{f(P(\Bx))}\\
&\le -f\p{\E_{\Bx \in X}\b{P(\Bx)}}\tag{convexity}\\
&= -f\p{\f1n},
}
\]
so the $f$-entropy takes a maximum of $-n f\p{\f1n}$ when $P$ is uniform.
Links to other quantities
Informally, $\sum_{x \in X} f(P(x))$ can be understood as the $f$-divergence $D^f\pff{P}{1}$, where $1$ is an imaginary prior distribution that puts probability $1$ on every point. In this sense, the $f$-entropy is (the negative of) the “absolute” version of the $f$-divergence, for a neutral prior. The closer $P$ is to uniform, the lower $D^f\pff{P}{1}$ is, and the higher the entropy $H^f\p P$ is.
More broadly, we’ll call “$f$-entropy” any (usually decreasing) function of $H^f\p P$. For example,
- the Shannon entropy is given by $\H(P) \ce H^f(P)$ for $f(p) \ce p \log p$ and ranges from $0$ (single point) to $\log n$ (uniform);
- the power entropies are given by $H_\alpha(P) \ce H^f(P)^{\f1{1-\alpha}}$ for $f(p) \ce p^\alpha$ and they all range from $1$ (single point) to $n$ (uniform).