Bartlett Identities

Definition

First Bartlett Identity

$0=E\left[\frac{\partial \ln f(X|\theta )}{\partial \theta }\right]=E[s(\theta |x)]$ where $f(X|\theta )$ is a Likelihood Function and $s(\theta |x)$ is a Score Function

Second Bartlett Identity

$0=E\left[\frac{{\partial }^2\ln f(X|\theta )}{\partial {\theta }^2}\right]+E\left[{\left(\frac{\partial \ln f(X|\theta )}{\partial \theta }\right)}^2\right]$ where $f(X|\theta )$ is a Likelihood Function