More on the partition function

The partition function is the generating function of expectation values, in particular there are mathematical relationships between expectation values and the log-partition function. In this case we have

$$\begin{align*} \langle \frac{ \partial E(\boldsymbol{x}; \Theta_i) } { \partial \Theta_i} \rangle_{model} = \int p(\boldsymbol{x}| \boldsymbol{\Theta}) \frac{ \partial E(\boldsymbol{x}; \Theta_i) } { \partial \Theta_i} d\boldsymbol{x} = -\frac{\partial \log Z(\Theta_i)}{ \partial \Theta_i} . \end{align*}$$

Here $\langle \cdot \rangle_{model}$ is the expectation value over the model probability distribution $p(\boldsymbol{x}| \boldsymbol{\Theta})$.

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