I am trying to prove that $f(\mu, \Sigma) = \left\langle\log \phi\left(w^{T} x\right)\right\rangle_{\mathcal{N}(x | \mu, \Sigma)} = \int p(x|\mu, \Sigma) \log \phi(w^Tx)dx$ is convex/concave in $(\mu, \Sigma)$ for some log-concave function $\phi(w^Tx)$.
Using Jensen's Inequality I managed to show this
\begin{align*} f(\alpha \mu_1 + (1-\alpha) \mu_2, \Sigma) &= \mathbb{E}_{x \sim N(\alpha \mu_1 + (1-\alpha) \mu_2, \Sigma)} [\log \phi(w^Tx)] \\ & \leq \log \phi(w^T (\alpha \mu_1 + (1-\alpha) \mu_2)) = \log \phi(\alpha w^T \mu_1 + (1-\alpha) w^T \mu_2)) \\ & \geq \alpha \log \phi(w^T\mu_1) + (1 - \alpha) \log \phi(w^T \mu_2) \\ & \geq \alpha f(\mu_1, \Sigma) + (1-\alpha) f(\mu_2, \Sigma) \end{align*}
However, it is only half of the proof. Another problem is convexity over $\Sigma$, because I do not see how e.g. Jensen's Inequality could help.
Thanks!