Definition · applies to statistical models · also called likelihood function

Undergraduate (statistics)

Consider a statistical model that gives a probability (or probability density) $p(x \mid \theta)$ of data $x$ for each value of a parameter $\theta$, and observed data $x$. The likelihood of $\theta$ is the probability of the observed data, viewed as a function of the parameter:

\[L(\theta) = p(x \mid \theta).\]

Properties

  • A likelihood is not a probability distribution over $\theta$: its values need not add up (or integrate) to $1$ over $\theta$.
  • The maximum likelihood estimate of $\theta$ is the value that maximizes $L(\theta)$.

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