Given an undirected graphical model with no edges and only N nodes, I am trying to find a closed form solution to the ML estimate of each node given that $p(x|\theta)=\frac{\exp(\sum_{s\in V}\theta_sx_s)}{\sum_x\exp(\sum_{s\in V}\theta_sx_s)}$, where $x=(x_1,\ldots,x_N)$ and the denominator is the partition function. I derived $\log p(x_1,\ldots,x_L|\theta)=\sum\limits_l\log p(x_l|\theta)=\sum\limits_s\theta_s\sum\limits_l x_{sl}-L\log\left(\sum\limits_x\exp\sum\limits_s\theta_s x_s\right)$, which is the log-likelihood function. To find the ML estimate of $\theta_s$, I tried to take the partial derivative wrt $\theta_s$ and set it to 0, but the partial derivative I get is $\sum\limits_lx_{sl}-\frac{L\sum\limits_xx_s\exp\sum\limits_s\theta_s x_s}{\sum\limits_x\exp\sum\limits_s\theta_s x_s}$ and at this point I don't know how to solve for $\theta_s$ and thus I can't get a closed form solution for the ML estimate. Can anyone help? Is my derivation correct? Am I even approaching this in the right way, or is there another approach I can take? Thanks!


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