Great question—this is one of those statistical terms that feels totally arbitrary at first, but makes perfect intuitive sense once you unpack its history and meaning.

First, let's recap the definition to ground us: the score function (the "score" you're asking about) is formally defined as:
$$U(\theta; x) = \frac{\partial}{\partial \theta} \log p(x; \theta)$$
where $p(x; \theta)$ is the likelihood of dataset $x$ given parameter $\theta$.

The Origin: Ronald Fisher's Exam Score Analogy #

This term was coined by the legendary statistician Ronald Fisher in the 1920s, and he chose "score" as a direct, relatable analogy to exam scores:

• Think of the log-likelihood as a "grading rubric" for how well a parameter $\theta$ fits the data $x$. The score function tells you how much the "grade" (log-likelihood) would change if you tweak $\theta$—it's like getting targeted feedback on an exam: "adjust this way to get a better score."
• When the score equals 0, you've hit the maximum of the log-likelihood (the maximum likelihood estimate, or MLE). This is equivalent to getting a perfect score: the parameter $\theta$ fits the data as well as possible.
• If the score is positive or negative, its sign tells you which direction to adjust $\theta$ to increase the likelihood, and its magnitude tells you how big that "grade boost" would be for a small tweak.

A Bonus Tie-In to Fisher Information #

The Fisher information matrix—another core concept from Fisher—is the variance of the score function. This reinforces the analogy: if the score varies a lot across different datasets, that means our "grading" of parameter $\theta$ is inconsistent, so we're less confident in our estimate (just like a test with high score variance is a less reliable measure of ability).

Fisher explicitly referred to this quantity as a measure of "fit quality" in his early work, and the "score" analogy stuck as the field adopted his framework.

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