📂Mathematical Statistics

Definition of Likelihood Function

Definition ¹

Let’s denote the joint probability density function or probability mass function of a sample $\mathbf{X} := \left( X_{1} , \cdots , X_{n} \right)$ as $f(\mathbf{x}|\theta)$. When a realization $\mathbf{x}$ is given, regarding $f(\mathbf{x}|\theta)$ as a function of $\theta$ $$ L \left( \theta | \mathbf{x} \right) := f \left( \mathbf{x} | \theta \right) $$ is called the Likelihood Function.

Explanation

In the context of discussing maximum likelihood estimators, it is necessary for the sample to be iid, but when discussing the Likelihood Principle, it is perfectly fine to consider the random vector itself without specifically thinking about random variables.

If for the parameter $\theta$, two parameters $\theta_{1}$ and $\theta_{2}$ $$ L \left( \theta_{1} | \mathbf{x} \right) \ge L \left( \theta_{2} | \mathbf{x} \right) $$ then it is said that $\theta_{1}$ is more Plausible than $\theta_{2}$ regarding $\theta$.