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Likelihood Function
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Definition of The Likelihood Function: In maximum likelihood estimation, the likelihood function (often denoted L()) is the joint probability function of the sample, given the probability distributions that are assumed for the errors. That function is constructed by multiplying the pdf of each of the data points together:
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In a case where the fit to the measurements is poor, perhaps because an inappropriate distribution function was used, the likelihood will have a value much smaller than expected. In that case, the estimates of uncertainty obtained from (4.6-4.8) should not be used. Instead, the proper conclusion is that the model used is inappropriate because it does not provide an adequate fit to the observations. Erroneously small estimates of uncertainty limits sometimes arise from using (4.6-4.8) when the fit is poor.
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The following figures show plots of likelihood, L, as a function of p for several different possible outcomes of n = 10 flips of a coin. Note that for the case in which 3
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Likelihood as a solitary term is a shorthand for [L]ikelihood function. In the colloquial language, "likelihood" is one of several informal synonyms for "probability", but throughout this article only the technical definition is used.
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This paper provides closed-form expansions for the transition density and likelihood function of arbitrary multivariate diffusions. The expansions are based on a Hermite series, whose coefficients are calculated explicitly by exploiting the special structure afforded by the diffusion hypothesis. Because the transition function for most diffusion models is not known explicitly, the expansions of this paper can help make maximum-likelihood a practical estimation method for discretely sampled multivariate diffusions. Examples of interest in financial econometrics are included.
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Likelihood::SpectrumFactory This class implements the Prototype pattern in order to provide a common point of access for retrieving and storing Functions for spectral modeling. Basic Functions such as Likelihood::PowerLaw, Likelihood::Gaussian, and Likelihood::AbsEdge are provided by default. Clients can combine models using the Likelihood::CompositeFunction hierarchy, store those models in SpectrumFactory, and later, clone those stored models for use in other contexts.
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Likelihood Function