As mentioned in the title, I am confused over the difference between $L(\theta|S)$ and $L(\theta|X)$, where $X = (X_1, X_2, ... ,X_n$).
From what I understand, $L(\theta|S)$ is the probability of $\theta$ being the true probability given a response $S$, which is the sample. This can be useful in a sense that the peak of $\theta$ can be shown through a graph. This peak represents the highest likelihood for the parameter $\theta$.
As for $L(\theta|X = X_1, X_2, ..., X_n)$, it implies the likelihood of $\theta$ given a set of different probability measures. This can be useful in the same way where an MLE can be derived using a set of steps (i.e. differentiating the score fn, etc.).
I'm not sure if I failed to understand the difference or if my textbook just didn't outline a difference. Or can they be used interchangeably?
p.s. In terms of using the peak of $\theta$, I'm trying to express my understanding of likelihood function, I know there are more ways to express it's usefulness.