Likelihood Principle

In statistics, the likelihood principle is a controversial principle of statistical inference which asserts that all of the information in a sample is contained in the likelihood function.

A likelihood function arises from a conditional probability distribution considered as a function of its distributional parameterization argument, conditioned on the data argument. For example, consider a model which gives the probability density function of observable random variable X as a function of a parameter θ. Then for a specific value x of X, the function L(θ | x) = P(X=x | θ) is a likelihood function of θ: it gives a measure of how "likely" any particular value of θ is, if we know that X has the value x. Two likelihood functions are equivalent if one is a scalar multiple of the other. The likelihood principle states that all information from the data relevant to inferences about the value of θ is found in the equivalence class. The strong likelihood principle applies this same criterion to cases such as sequential experiments where the sample of data that is available results from applying a stopping rule to the observations earlier in the experiment.