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 $L(\theta) = \ln(Z(\theta)) - q^{T} M\theta$. We denote $$\Sigma_{N}(\bar{g}, g) = \left\{p \in \Sigma_{N} \mid \sum_{i=1}^{N} p_{i} g_{k}(i) = \bar{g}_{k}, 1 \leqslant k \leqslant d\right\}.$$ Show that if $\theta$ is a critical point of $L$ (that is, a point where the gradient of $L$ vanishes) then $M^{T} p(\theta) = M^{T} q$ and $p(\theta) \in \Sigma_{N}(\bar{g}, g)$.
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 $L(\theta) = \ln(Z(\theta)) - q^{T} M\theta$. We denote
$$\Sigma_{N}(\bar{g}, g) = \left\{p \in \Sigma_{N} \mid \sum_{i=1}^{N} p_{i} g_{k}(i) = \bar{g}_{k}, 1 \leqslant k \leqslant d\right\}.$$
Show that if $\theta$ is a critical point of $L$ (that is, a point where the gradient of $L$ vanishes) then $M^{T} p(\theta) = M^{T} q$ and $p(\theta) \in \Sigma_{N}(\bar{g}, g)$.