I know that the Fisher Information is defined as the variance of the score function: $$ I(\theta)=Var(\frac{d}{d\theta}\mathrm{log}L(x|\theta))=\int(\frac{d}{d\theta}\mathrm{log}f(x|\theta))^2p_\theta(x)dx,\;\;\;\;\mathrm{with}\;p_\theta(x)= f(x|\theta) $$ where $\mathrm{log}L(x|\theta)$ is the log-likelihood. Under "mild regularity conditions" this can also be written as: $$ I(\theta)=-E_\theta(\frac{d^2}{d\theta^2}\mathrm{log}f(x|\theta))=-\int(\frac{d}{d\theta}\mathrm{log}f(x|\theta))^2p_\theta(x)dx $$ I have an intuition for the second definition. However, I am not sure why the first definition works and how the two are connected.
What am I missing?
