Likelihooden

Likelihooden is a generalized concept used in statistics to denote an object that aggregates or summarizes likelihood information across a family of models or parameterizations. It serves as a flexible framework for comparing competing explanations of data by extending the traditional notion of a single likelihood function.

Formally, for data x and a set of models M with parameters θ in Θ_M, each model has a likelihood L(θ; x) = p(x | θ, M). The likelihooden can be defined in two common ways: as a vector-valued collection {L(θ; x) : θ ∈ Θ_M, M ∈ scriptM}, which preserves parameter- and model-specific evidence, or as a scalar summarization for each model class, such as L*(x | M) = sup_{θ ∈ Θ_M} L(θ; x) or p(x | M) = ∫ L(θ; x) π_M(θ) dθ, the latter being the marginal likelihood. The likelihooden across models can then be represented as the set {L*(x | M_i), p(x | M_i)} for all models M_i in the considered family.

Uses and interpretation: Likelihooden provides a unified way to compare models by their fit to the data, while accounting for model complexity through chosen scalars like maximum likelihood or marginal likelihood. It relates to traditional tools such as likelihood ratios, information criteria, and Bayesian evidence, but emphasizes a structured aggregation of evidence across models or parameter regimes.

Computation and considerations: Evaluating the likelihooden often requires optimization over parameters and, for marginal likelihoods, integration or approximation techniques (e.g., Laplace methods, variational inference). As with any model comparison, sensitivity to priors, parameterization, and data size should be considered.

See also: likelihood, likelihood ratio, Bayesian model comparison, marginal likelihood, information criteria.