Likelihood Function

In statistical theory, the likelihood function is a fundamental concept used for parameter estimation. It quantifies the probability of observing the given data under specific parameter values of a statistical model. Formally, if we denote the observed data as X and the parameters as θ, the likelihood function L(θ | X) is defined as L(θ | X) = P(X | θ), where P(X | θ) is the probability of the data X given the parameters θ. This function is crucial in the context of maximum likelihood estimation (MLE), where the goal is to find the parameter values that maximize the likelihood function. The likelihood function is closely related to the concept of probability density functions in continuous distributions and probability mass functions in discrete distributions. It serves as the foundation for various statistical inference techniques, including hypothesis testing and Bayesian inference, where it is combined with prior distributions to form posterior distributions via Bayes' theorem. The mathematical properties of the likelihood function, such as its concavity and differentiability, are essential for deriving estimators and conducting statistical tests.