In order to simplify notation we defined the "operator"
$$ \begin{align*} O_i(\boldsymbol{x}) = \frac{\partial E(\boldsymbol{x}; \Theta_i)}{\partial \Theta_i}, \end{align*} $$and used the statistical mechanics relationship between expectation values and the log-partition function:
$$ \begin{align*} \langle O_i(\boldsymbol{x}) \rangle_{model} = \text{Tr} P_\Theta(\boldsymbol{x})O_i(\boldsymbol{x}) = - \frac{\partial \text{log} Z(\{ \Theta_i\})}{\partial \Theta_i}. \end{align*} $$