I would like to understand how to calculate the Hessian of $\mathbb{E}_{\xi}[\log(1+e^{a^\top (x+\xi)})]$ with respect to $a$. Say $\xi\sim \mathcal{N}(0, \sigma^2I)$, $a$ and $x$ are both fixed.
In the one dimensional case, it seems I can exchange the order of differentiation and integration using the Leibniz integration rule: $$ \frac{d^2}{da^2}\mathbb{E}_{\xi}[\log(1+e^{a^\top (x+\xi)})] =\mathbb E_{\xi}\left[ \frac{(x+\xi)^2e^{a(x+\xi)}}{(1+e^{a(x+\xi)})^2} \right] $$ Even so I'm still not sure whether I can integrate this.
Does this Hessian have a closed form expression? Thanks!