Likelihoodbased

Likelihood-based refers to statistical methods that use the likelihood function as the foundation for estimation, inference, and model comparison. The likelihood function, L(θ; data), expresses the probability of observing the data given a parameter value θ. The central goal is to learn about θ from the observed data by focusing on how likely different parameter values are.

Estimation in likelihood-based approaches typically uses maximum likelihood estimation (MLE), selecting θ̂ that maximizes the likelihood. Under regularity conditions, θ̂ is consistent and asymptotically normal, with an estimated variance given by the inverse Fisher information. Inference often employs likelihood-based procedures such as the likelihood ratio test, the Wald test, or score tests. The likelihood ratio test compares the maximum likelihood under a full model to that under a constrained model; by Wilks’ theorem, the statistic -2 log λ follows an asymptotic chi-square distribution. Confidence intervals can be formed from the profile likelihood, which fixes a parameter of interest and maximizes over the remaining parameters.

Extensions include information criteria for model selection (AIC, BIC) derived from the likelihood, as well as semi-parametric and high-dimensional settings, composite likelihoods, and pseudo-likelihoods when the full likelihood is intractable. Limitations include sensitivity to model misspecification and the need for regularity conditions; computations can be intensive in complex models. While often contrasted with Bayesian methods, likelihood-based approaches share the same likelihood function but differ in interpretation and incorporation of prior information.