Likelihood function

Builds on Probability mass function Probability density function Independent and identically distributed random variables

Definition

Let $\Theta$ be a set of parameters. For each $\theta \in \Theta$ , let $f(\cdot \mid \theta)$ be a probability mass function or a density on $\mathbb{R}^n$ . It gives the law of the data $X = (X_1, \dots, X_n)$ when $\theta$ is the true parameter. For observed data $x \in \mathbb{R}^n$ , the likelihood function is the function $L \colon \Theta \to [0, \infty)$ with $L(\theta \mid x) = f(x \mid \theta).$ The log-likelihood is $\ell(\theta \mid x) = \ln L(\theta \mid x)$ .

Remarks

For an i.i.d. sample with mass function or density $f_1(\cdot \mid \theta)$ for each observation, $L(\theta \mid x) = \prod_{i=1}^{n} f_1(x_i \mid \theta)$ and $\ell(\theta \mid x) = \sum_{i=1}^{n} \ln f_1(x_i \mid \theta)$ . The likelihood is not a probability distribution on $\Theta$ . In general, its integral over $\theta$ is not $1$ . Example: for $k$ successes in $n$ Bernoulli( $p$ ) trials, $L(p \mid x) = p^k (1 - p)^{n - k}$ for $p \in [0, 1]$ . Only ratios $L(\theta_1 \mid x) / L(\theta_2 \mid x)$ matter for inference. These ratios compare how well two parameter values explain the data.

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Independent and identically distributed random variables definition
Likelihood function definition

Tag	`05FSTXM92W`
URI	`m2://05FSTXM92W`
Kind	definition
Writing	W1 stated
MSC	62F10
Defines	Likelihood function
Uses	Probability mass function, Probability density function, Independent and identically distributed random variables

Likelihood function

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