Maximum Likelihood Estimation

Setup

We have a random variable $X$ which can be parameterised using $\theta$, where $X$ and $\theta$ might be scalars, vectors, etc. We have a probability distribution function $f(x,\theta )$ which describes the probability of making the observation $x\in X$ for a specific parameter $\theta$, i.e. $\mathbb{P}[X\in A]={\int }_{A}f(x;\theta )dx$.

Our goal is to find the parameters $\theta$ that maximise our observations $\{x_i{\}}_i$.

Procedure

1. We define the likelihood of our observations $L=L_{\theta }(\{x_i{\}}_i)={\prod }_{i}f(x_i,\theta )$.
2. We calculate, usually analytically, $\underset{\theta }{\mathrm{argmax}}\left\{L_{\theta }(\{x_i{\}}_i)\right\}=\underset{\theta }{\mathrm{argmax}}\left\{{\prod }_{i}f(x_i,\theta )\right\}$.
   1. Since $argmax$ is constant under Monotonic transformations, it is often useful to instead calculate $\underset{\theta }{\mathrm{argmax}}\left\{{\prod }_{i}f(x_i,\theta )\right\}=\underset{\theta }{\mathrm{argmax}}\left\{\log \left({\prod }_{i}f(x_i,\theta )\right)\right\}=\underset{\theta }{\mathrm{argmax}}\left\{{\sum }_i\log \left(f(x_i,\theta )\right)\right\}$

Even though the procedure above finds us the $\theta$ that maximises the likelihood, it does not provide us with information about the uncertainty around $\theta$. This could be measured with the Fisher Information. I am not familiar with other standardised methods of obtaining such a result. Help me ❓