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. This principle is controversial because it is inconsistent with the mainstream frequentist approach to inference. While the likelihood function is important to frequentists, they do not accept the likelihood principle. 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 f X ( x ∣ θ ) {\displaystyle \;f_{X}(x\mid \theta )\;} of observable random variable X {\displaystyle \,X\,} as a function of a parameter θ {\displaystyle \,\theta ~} . Then for a specific value x {\displaystyle \,x\,} of X {\displaystyle \,X~} , the function L ( θ ∣ x ) = f X ( x ∣ θ ) {\displaystyle \,{\mathcal {L}}(\theta \mid x)=f_{X}(x\mid \theta )\;} is a likelihood function of θ {\displaystyle \,\theta ~} : it gives a measure of how "likely" any particular value of θ {\displaystyle \,\theta \,} is, if we know that X {\displaystyle \,X\,} has the value x {\displaystyle \,x~} . 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.
Example Suppose
X {\displaystyle \ X\ } is the number of successes in twelve independent Bernoulli trials with each attempt having probability θ {\displaystyle \ \theta \ } of success on each trial, and
Y {\displaystyle \ Y\ } is the number of independent Bernoulli trials needed to get a total of three successes, again each attempt with probability θ {\displaystyle \ \theta \ } of success on each trial (if it was a fair coin each toss would have θ = 1 2 {\displaystyle \ \theta ={\tfrac {\!\ 1\!\ }{2}}\ } of either outcome, heads or tails). Then the observation that X = 3 {\displaystyle \ X=3\ } induces the likelihood function
L ( θ ∣ X = 3 ) = ( 12 3 ) θ 3 ( 1 − θ ) 9 = 220 θ 3 ( 1 − θ ) 9 , {\displaystyle \ \operatorname {\mathcal {L}} \left(\ \theta \ \mid \ X=3\ \right)={\binom {12}{3}}~\theta ^{3}\ (1-\theta )^{9}=220\ \theta ^{3}\ (1-\theta )^{9}\ ,}
while the observation that Y = 12 {\displaystyle \ Y=12\ } induces the likelihood function
L ( θ ∣ Y = 12 ) = ( 11 2 ) θ 3 ( 1 − θ ) 9 = 55 θ 3 ( 1 − θ ) 9 . {\displaystyle \ \operatorname {\mathcal {L}} \left(\ \theta \ \mid \ Y=12\ \right)={\binom {11}{2}}~\theta ^{3}\ (1-\theta )^{9}=55\ \theta ^{3}\ (1-\theta )^{9}~.}
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