Likelihood principle

In statistics, the likelihood principle is the proposition that, given a statistical model, all the evidence in a sample relevant to model parameters is contained in the likelihood function. A likelihood function arises from a probability density function considered as a function of its distributional parameterization argument. For example, consider a model which gives the probability density function of observable random variable as a function of a parameter Then for a specific value of the function is a likelihood function of it gives a measure of how "likely" any particular value of is, if we know that has the value The density function may be a density with respect to counting measure, i.e. a probability mass function. Two likelihood functions are equivalent if one is a scalar multiple of the other. The likelihood principle is this: All information from the data that is relevant to inferences about the value of the model parameters is in the equivalence class to which the likelihood function belongs. 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. Suppose is the number of successes in twelve independent Bernoulli trials with probability of success on each trial, and is the number of independent Bernoulli trials needed to get three successes, again with probability of success on each trial ( for the toss of a fair coin). Then the observation that induces the likelihood function while the observation that induces the likelihood function The likelihood principle says that, as the data are the same in both cases, the inferences drawn about the value of should also be the same. In addition, all the inferential content in the data about the value of is contained in the two likelihoods, and is the same if they are proportional to one another.