Definition
F(\theta) = g_{ij}(\theta)
&:= E_{\theta}\left[ \frac{\partial\ln L(\theta|x)}{\partial \theta_{i}}\frac{\partial\ln L(\theta|x)}{\partial \theta_{j}} \right]
= \int_{\mathbb{R}} \frac{\partial\ln L(\theta|x)}{\partial \theta_{i}}\frac{\partial\ln L(\theta|x)}{\partial \theta_{j}} p(x, \theta)dx\\
&= -E_{\theta}\left[ \frac{\partial^{2}\ln L(\theta|x)}{\partial \theta_{i}\theta_{j}} \right]
= -\int_{\mathbb{R}} \frac{\partial^{2} \ln L(\theta|x)}{\partial \theta_{i}\theta_{j}} p(x, \theta)dx
\end{aligned}$$
by the [[Bartlett Identities#second-bartlett-identity|second Bartlett identity]]
# Facts
## Monotonicity of Fisher Information Metrics
> Suppose a [[Probability Space]] $(\Omega, {\cal F}, P)$, a [[Random Variable]] $X: \Omega \rightarrow \mathbb{R}$, and Borel measurable function $T: \mathbb{R} \to \mathbb{R}$
> Then, the fisher information metrics of two random variables $X, T\circ X$ hold relation:
> $$I_{T \circ X}(\theta) \leq I_{X}(\theta)$$
> In other words, $\forall v \in \mathbb{R}_{n}, v^{\intercal}I_{T \circ X}(\theta)v \leq v^{\intercal}I_{X}(\theta)v$
> where $I_{X}(\theta) := E_{\theta}\left[ \left( \frac{\partial\ln f_{\theta}}{\partial \theta} \right)^{2} \right]$, $I_{T\circ X}(\theta) := E_{\theta}\left[ \left( \frac{\partial\ln \phi_{\theta}}{\partial \theta} \right)^{2} \right]$ and $f_{\theta}, \phi_{\theta}$ are the [[Density Function#induced-dessity-function|induced density functions]] by the $X$ and $T\circ X$ respectively.
> Equality holds if and only if $T$ is [[Sufficient Statistic]]
> $$E\left[ \frac{d \ln f_{\theta}}{d\theta}|T=t \right] = \frac{d\ln\phi_\theta(t)}{d\theta}$$