Functional derivative of differential entropy I have trouble finding the derivative of the differential entropy w.r.t the probability density function, i.e. what is  $\frac{\delta F[p]}{\delta p(x)}$, where $F[p] = \int_X p(x)\ln(p(x))dx$, and $p(x)$ is a probability density function.
I tried finding the derivative with the definition of the Frechet derivative, but was not able to solve it. Also, there already exist many posts on the derivative of the related Shannon Entropy, but I was not able to find anything on the differential entropy.
Any ideas?
Thanks,
Jan
 A: Looking at your "variational" $\delta$-notation, it appears that it will be enough for you to consider the Gateaux derivative, which is a weaker version of the Fréchet derivative. For $F(p):=\int_X p\ln p\,dx$, the Gateaux derivative of $F$ at a point $p$ in the direction $h\in L^1(X)$ is $f'(0)$, where
$$f(t):=F(p+th).$$
For variational purposes, it would be even better to consider just the (right) directional derivative of $F$ at point $p$ in the direction $h\in L^1(X)$:
$$(D_+F)(p)(h):=f'_+(0):=\lim_{t\downarrow0}\frac{F(p+th)-F(p)}t$$
whenever this limit exists.
Even this right directional derivative will not in general exist for all $h\in L^1(X)$; e.g., it will not exist if the set on which $h<0$ and $p=0$ is of nonzero measure, because on that set $(p+th)\ln(p+th)$ will not be even be defined for any real $t>0$. However, noting that $(u\ln u)'=1+\ln u$ for real $u>0$, we see that, by the dominated convergence theorem, $(D_+F)(p)(h)$ will exist and be given by
the formula
$$(D_+F)(p)(h)=\int_X (1+\ln p)h\,dx$$
for any $h\in L^1(X)$ such that $|h\ln(p+th)|\le g$ for some $g\in L^1(X)$ and all small enough $t>0$.
