Maximum likelihood estimation solves $\hat\theta_{\text{MLE}}=\arg\max_{\theta\in\Theta} \ell(\theta)$, where $\ell(\theta)=\log L(\theta;x)$. Under regularity conditions, MLEs are consistent, asymptotically normal, and asymptotically efficient.

The score is $U(\theta)=\partial \ell(\theta)/\partial \theta$. The expected Fisher information is $I(\theta)=\mathbb{E}\left[-\partial^2 \ell(\theta)/\partial \theta^2\right]$. In regular models, the Cramér-Rao lower bound states that for unbiased estimators $\mathrm{Var}(\hat\theta)\ge I(\theta)^{-1}$.

A full exponential family has density $f(x\mid \eta)=h(x)\exp\{\eta^T T(x)-A(\eta)\}$. Here $T(x)$ is a sufficient statistic, $\eta$ is the natural parameter, and $A(\eta)$ is the log-partition function.

Information geometry interprets statistical models as manifolds equipped with the Fisher metric. This perspective helps explain efficiency, dual connections, and the geometry of estimation and divergence minimization.