TITLE: Intuitive explanation of a definition of the Fisher information
QUESTION [72 upvotes]: I'm studying statistics. When I read the textbook about Fisher Information, I couldn't understand why the Fisher Information is defined like this:
$$I(\theta)=E_\theta\left[-\frac{\partial^2 }{\partial \theta^2}\ln P(\theta;X)\right].$$
Could anyone please give an intuitive explanation of the definition?

REPLY [5 votes]: All these are correct, but they do not explain why we need to look at the curvature (Hessian) of the log-likelihood instead of the likelihood. 
Put very informally: asymptotic normality states that the distribution of MLE estimators around mode is close to the likelihood (or mimics curvature of the likelihood) as number of samples approaches to infinity. The shape of the distribution gets close to a normal distribution centered on that mode and has the same curvature as the likelihood (NOT log-likelihood) at the mode.
I think the ideas behind asymptotic normality and Laplace approximation are intimately related. It is almost similar to an Laplace approximation around the mode of the likelihood.