Let $M \in \mathscr{M}_{N,d}(\mathbb{R})$ defined by $M_{i,j} = g_{j}(i)$, $q \in \Sigma_{N}$, and for all $\theta \in \mathbb{R}^{d}$, $f(\theta) = M\theta$, $Z(\theta) = \sum_{i=1}^{N} e^{f_{i}(\theta)}$, $$p(\theta) = \left(\frac{e^{f_{1}(\theta)}}{Z(\theta)}, \ldots, \frac{e^{f_{N}(\theta)}}{Z(\theta)}\right) \in \Sigma_{N},$$ and $L(\theta) = \ln(Z(\theta)) - q^{T} M\theta$. Show that $L$ is of class $\mathscr{C}^{2}$ and that for all integers $1 \leqslant l, k \leqslant d$ we have $$\frac{\partial^{2} L}{\partial \theta_{l} \partial \theta_{k}}(\theta) = \sum_{i=1}^{N} p_{i}(\theta)(M_{il} - m_{l}(\theta))(M_{ik} - m_{k}(\theta))$$ where $m(\theta) = M^{T} p(\theta)$.
Let $M \in \mathscr{M}_{N,d}(\mathbb{R})$ defined by $M_{i,j} = g_{j}(i)$, $q \in \Sigma_{N}$, and for all $\theta \in \mathbb{R}^{d}$, $f(\theta) = M\theta$, $Z(\theta) = \sum_{i=1}^{N} e^{f_{i}(\theta)}$,
$$p(\theta) = \left(\frac{e^{f_{1}(\theta)}}{Z(\theta)}, \ldots, \frac{e^{f_{N}(\theta)}}{Z(\theta)}\right) \in \Sigma_{N},$$
and $L(\theta) = \ln(Z(\theta)) - q^{T} M\theta$.
Show that $L$ is of class $\mathscr{C}^{2}$ and that for all integers $1 \leqslant l, k \leqslant d$ we have
$$\frac{\partial^{2} L}{\partial \theta_{l} \partial \theta_{k}}(\theta) = \sum_{i=1}^{N} p_{i}(\theta)(M_{il} - m_{l}(\theta))(M_{ik} - m_{k}(\theta))$$
where $m(\theta) = M^{T} p(\theta)$.