The Fisher information is defined in two equivalent ways: as the variance of the slope of $\ell(x)$, and as the negative of the expected curvature of $\ell(x)$. Since the former is always positive, this would imply that the curvature of the log-liklihood function is everywhere negative. This seems plausible to me, since every distribution that I have seen has a log-likelihood function with negative curvature, but I don't see why this must be the case.
Solved – Is it the case that the log-likelihood *always* has negative curvature? Why
fisher informationlikelihoodmaximum likelihood
Best Answer
Your conclusion doesn't follow: if the expected value of the curvature of the log-likelihood is negative, it is not necessarily everywhere negative. It just needs to be, on average, more negative than positive. Think of a bimodal distribution: there is indeed a region in between the modes with positively curved log-likelihood, so your claim cannot be true.
Note the link with maximum likelihood estimation for intuition: in the neighborhood of the MLE, you may expect the curvature to be negative because you are at a maximum (although it is not necessarily, like if the maximum occurs on the boundary, for example). If the curvature is negative in the most likely regions, then the average should tend to be negative, intuitively. In fact, it must always be, under the regularity conditions that allow you to use the equivalency with the "variance of the slope" definition, as you point out.