score function

Consider the log-likelihood for a parameter $\theta$:

$$\ell (\theta )=\log \mathcal{L}(\theta )=\log \mathbb{P}(X^n|\theta ).$$

The score function is the gradient of $\ell (\theta )$. The score function is used to define the Fisher information. Note that the score is data-dependent — it is a random quantity. It also comes up when analyzing the asymptotics of the MLE.