Likelihood function

Definition: A statistical model of the data used in frequentist and Bayesian analyses, defined up to a constant of proportionality. A likelihood function represents the likeliness of different parameters for your distribution given the data. Given that probability distributions have unknown population parameters, the likelihood function indicates how well the sample data summarise these parameters. As such, the likelihood function gives an idea of the goodness of fit of a model to the sample data for a given set of values of the unknown population parameters.

Alternative definition: For a more statistically-informed definition, given a parametric model specified by a probability (densidity) function f(x|theta), a likelihood for a statistical model is defined by the same formula as the density except that the roles of the data x and the parameter theta are interchanged, and thus the likelihood can be considered a function of theta for fixed data x. Here, then, the likelihood function would describe a curve or hypersurface whose peak, if it exists, represents the combination of model parameter values that maximize the probability of drawing the sample obtained.

Related terms: Bayes factor, Bayesian inference, Bayesian parameter estimation, Posterior distribution, Prior distribution

References: Dienes (2008), Hogg et al. (2010), van de Schoot et al. (2021), Geyer (2003), Geyer (2007), & https://blog.stata.com/2016/11/01/introduction-to-bayesian-statistics-part-1-the-basic-concepts/

Drafted and Reviewed by: Alaa AlDoh, Dominik Kiersz, Graham Reid, Sam Parsons, Flávio Azevedo

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