# Instead of computing Fisher information, why don't we just evaluate the second derivative of log(L) at the mle?

So I (sort of) understand that Fisher information is $$I(\theta) = -E_{\theta}[\frac{d^2}{d\theta^2} log f(x|\theta)]$$, but what I'm confused by is why we bother taking the expectation with respect to $$\theta$$, instead of just evaluating $$\frac{d^2}{d\theta^2} log f(x|\theta)$$ at $$\theta = \hat{\theta}_{mle}$$.

Wouldn't this point evaluation give us a much better sense of how sharply peaked the log-likelihood is at its maximum, rather than taking the whole expectation? I worry that we can have a situation where there are many regions $$\theta =/=\hat{\theta}_{mle}$$ where the second derivative is very large, which could obsure curvature being very low at $$\theta=\hat{\theta}_{mle}$$.

• The expectation is taken over $x$ not $\theta$ - that's why the result is a function of $\theta.$ – Dap Feb 26 at 14:41
• Ah, okay. That makes way more sense. Thanks. – user49404 Feb 26 at 15:00