# Is it the case that the log-likelihood *always* has negative curvature? Why?

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.