Likelihood Function

Definition

Definition. Let $X$ be random variable with probability distribution $f:{\mathbb{R}}^n\times {\mathbb{R}}^m\to \mathbb{R}$ depending on a parameters $\theta$. Then, for a given outcome $x\in {\mathbb{R}}^n$, the function

$$\begin{array}{rcll}L:& \{x\}\times {\mathbb{R}}^m& \to & \mathbb{R}\\ & (x,\theta )& \mapsto & L(\theta |x)=f(x|\theta )\end{array}$$

is said to be the likelihood function of $\theta$.

Remarks

Note that, $L(\theta |x)$ does not specify the probability that $\theta$ is true, given the observed outcome $x$.