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Let $X_1,...,X_n$ be some observations, and let $\theta$ be some parameter of the density function we want to estimate.

Then, it is well known that

$l(\theta) = n^{-1} \sum_{i=1}^n \log f(X_i ; \theta)$ is called the average log-likelihood.

What is $\mathbb{E}[\log f(X_1 ; \theta)]$ called? Meaning, "the expected value of the likelihood"? It can be called the cross-entropy, or perhaps it has other name, but it seems to me it should have a name relating it to the likelihood, such as "population likelihood" perhaps or something of that sort.

Anyone knows? Wikipedia did not help much here.

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    I know that the second derivative is called [Fisher information](http://en.wikipedia.org/wiki/Fisher_information). I don't recall a special name for the expectation of the log-likelihood itself.2010-12-30
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    what is the expectation over? and why is there $X_1$ and not $X$ in there? Otherwise, isn't this expectation just the negative of the entropy?2010-12-30

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