The derivative of the log-likelihood is known as the score:
\[\u(\bt) = \nabla \ell(\bt \mid \x).\]The score, along with the information, is a critical component in obtaining quadratic approximations to log-likelihood functions.
Note that
- The score $\u$ is a function of $\theta$.
- For any given $\bt$, $\u(\bt)$ is a random variable, as it depends on the data $\x$; usually suppressed in notation.
- For independent observations, the score of the entire sample is the sum of the scores for the individual observations:
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By the chain rule,
\[\u(\bt) = \frac{\nabla_{\bt} p(x \mid \bt)}{p(x \mid \bt)},\]where $p$ is the density function.
Score equations
If the likelihood is regular, we can find $\bth$ by setting the gradient equal to zero; the MLE is the solution to the equation(s)
\[\u(\bt) = \zero;\]this system of equations is known as the score equation(s) or sometimes the likelihood equation(s).