Clarifying the definition of Fisher information

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:

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!

• 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 $\theta$. May 5 '20 at 6:05
• @TimMak Ok, so do you mean that $f(X;\theta)=f(x_1,\ldots,x_n|\theta)$?
– 李智修
May 5 '20 at 6:14
• 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 $\theta$. May 5 '20 at 8:26
• Ok, I now understand it. Great thanks!
– 李智修
May 10 '20 at 11:05

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 $$f(X;\theta)$$.