Likelihood

Likelihood is a concept in statistics and probability theory that measures the goodness of fit of a statistical model to a sample of data for given values of the unknown parameters. It is used in the method of maximum likelihood estimation, which seeks the parameter values that maximize the likelihood.

Definition

In the context of a statistical model, the likelihood of a set of parameter values, given some observed outcomes, is equal to the probability of those observed outcomes given those parameter values. More specifically, if the statistical model defines a distribution over outcomes, indexed by the parameters, then the likelihood of the parameters is defined as the probability of the observed outcomes under that distribution.

Properties

The likelihood has several important properties that make it useful for statistical inference. These include:

• Invariance to reparameterization: The likelihood remains the same under one-to-one transformations of the parameter space.
• Sufficiency: If a statistic is sufficient for the parameter, then the likelihood depends on the data only through that statistic.
• Factorization: If the data are independent given the parameter, then the likelihood factors into a product of individual likelihoods.

Applications

The likelihood is used in many areas of statistics, including:

• Maximum likelihood estimation: This is a method of estimating the parameters of a statistical model by maximizing the likelihood.
• Likelihood-ratio test: This is a statistical test based on the ratio of the likelihoods under the null and alternative hypotheses.
• Bayesian inference: In this context, the likelihood is used to update the prior distribution on the parameters to obtain the posterior distribution.

See also

• Probability density function
• Statistical model
• Bayesian statistics
• Maximum likelihood

References

Likelihood Resources
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