I'm studying the Fisher information, which leads me to the wiki page:
Formally, the partial derivative with respect to θ of the natural logarithm of the likelihood function is called the “score”. The variance of the score is defined to be the Fisher information. If log f(x; θ) is twice differentiable with respect to θ, and under certain regularity conditions, then the Fisher information may also be written as:
However, in the very beginning, the notation was specified as follows:
Let f(X; θ) be the probability density function (or probability mass function) for X conditional on the value of θ.
Hence, it makes me comes to a question. Is the f in the picture a density function or a likelihood function?
I've checked the wiki page of the likelihood function and they are obviously different. https://en.wikipedia.org/wiki/Likelihood_function
and for a statistic like $X_1,\ldots,X_n$, the likelihood function should be something like: $f(x_1,\ldots,x_n|\theta)$ right?
Which definition should I take in? or have I taken it totally wrong at some point? lol