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I'm studying the Fisher information, which leads me to the wiki page:

https://en.wikipedia.org/wiki/Fisher_information

It says:

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: Fisher information

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

Great thanks!

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    $\begingroup$ A likelihood function is a pdf (or a pmf). The reason why people often use ";" instead of "|" in a likelihood is that it is often taken as a function on $\theta$. Note it is a function on $\theta$, but not a pdf on $\theta$. $\endgroup$
    – Tim Mak
    May 5 '20 at 6:05
  • $\begingroup$ @TimMak Ok, so do you mean that $f(X;\theta)=f(x_1,\ldots,x_n|\theta)$? $\endgroup$
    – 李智修
    May 5 '20 at 6:14
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    $\begingroup$ Yes. The whole problem is that of notation. $f()$ is often used to denote a pdf. However, it is also a general notation for a function. The $f(x;\theta)$ notation allows it to be both a pdf on $x$ and a function on $\theta$. $\endgroup$
    – Tim Mak
    May 5 '20 at 8:26
  • $\begingroup$ Ok, I now understand it. Great thanks! $\endgroup$
    – 李智修
    May 10 '20 at 11:05
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I would think about your goal: when you use a probability density function (pdf) you are computing information on $X$ based on some known "fixed" parameters $\theta$. On the other side, you use the likelihood to get information about some parameters $\theta$ based on a sample that you observed $x = (x_1, x_2, \dots x_n)$, so it is a function of $\theta$ for an observed "fixed" sample $x$.

So in this sense you can use the notation $f(X|\theta)$ or $f(\theta|x)$ for $f(X;\theta)$.

In any case remember that the pdf is not a probability.

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