Likelihood function

Likelihood function is a fundamental concept in statistical inference, particularly in maximum likelihood estimation. It is a function of the parameters of a statistical model, given specific observed data. The likelihood of a set of parameter values, given outcomes x, is equal to the probability of those observed outcomes given those parameter values.

Definition

In the context of a statistical model, a likelihood function (often simply the likelihood) measures the goodness of fit of a statistical model to a sample of data for given values of the unknown parameters. It is formed from the joint probability distribution of the sample, but viewed and used as a function of the parameters only, thus treating the random variables as fixed at the observed values.

Properties

The likelihood function has several important properties that can be used in the process of parameter estimation. These include invariance to one-to-one transformations and factorization properties.

Applications

The likelihood function is used in a variety of statistical techniques including maximum likelihood estimation, likelihood-ratio test, and Bayesian inference.

See also

• Probability density function
• Probability mass function
• Sufficiency (statistics)
• Neyman–Pearson lemma

References

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  Likelihood function after observing HH

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  Likelihood function after observing HHT