The $\phi$ entropy is defined as $\text{Ent}\_{\phi}[X]= \mathbb{E}[\phi (X)]-\phi(\mathbb{E}[ X])$ where $X$ is a random variabel and $\phi$ is a convex function ($\text{Ent}\_{\phi}[X] \geq 0$). By choosing $\phi(x)=x^2$ we get $\text{Ent}\_{\phi}[X]=\text{Var}(X)$. if $\phi(x)=x\log x$ and $X=\frac{dv}{d\mu} $ (radom nikodym derivative) we get $\text{Ent}\_{\phi}[X]= D\_{\text{KL}}(v||\mu)$. Can we recover the classical entropy $H(X)= -\mathbb{E}[\log p(X)]$ in a similar way?


[Edit: added a minus sign in the definition of $H(X)$ -AK]



$\phi$ - Entropies - $\phi$ - Divergences and classical entropy recovery
The choice $\phi(x)=x\log(x)$ yields an entropy distinct from Shannon entropy. See p. 94 of Concentration Inequalities: A Nonasymptotic Theory of Independence by Boucheron, Lugosi, Massart:
<http://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780199535255.001.0001/acprof-9780199535255>


The two entropies are related but distinct.


