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Studying asymptotics, I bumped into the concept of Observed Fisher Information, as a way to compute Fisher Information when the parameter $\theta$ is unknown. I am also aware that it is related in some ways to Maximum Likelihood Estimators but I am a bit confused.

Thus my questions are: Which is the difference between Expected and Observed Fisher Information? How the observed Fisher information is computed? And how it is used?

EDIT: From the linked pages, I have understood how the Observed Fisher Information is computed and how it may be better than the Expected one in cases of finite samples. But I have some doubts yet: is the only difference between Observed and Expected Information the presence of an estimate (possibly the MLE of $\theta$) or a known $\theta$ ? Thus, should it be used, for example, in the case of asymptotic estimations?

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    $\begingroup$ Hi Alessandro. My answer here contains the gist of it. Maybe this helps you to get an idea. $\endgroup$ – COOLSerdash Dec 27 '15 at 8:30
  • $\begingroup$ Thank you! The derivation as well as the computations are really well explained. Have you also any hint about the use of Obs Fisher Info? $\endgroup$ – PhDing Dec 27 '15 at 8:50
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    $\begingroup$ Whenever you calculate the Fisher inforamtion using a MLE estimate of the parameter, you are calculating the observed Fisher information. It's called observed, because the MLE depends of the observed data. Here is an article that explains the differences and here is another, similar post on this site. $\endgroup$ – COOLSerdash Dec 27 '15 at 9:14

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