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 ${\displaystyle \;f_{X}(x\mid \theta )\;}$ of observable random variable ${\displaystyle \,X\,}$ as a function of a parameter ${\displaystyle \,\theta ~}$. Then for a specific value ${\displaystyle \,x\,}$ of ${\displaystyle \,X~}$, the function ${\displaystyle \,{\mathcal {L}}(\theta \mid x)=f_{X}(x\mid \theta )\;}$ is a likelihood function of ${\displaystyle \,\theta ~}$: it gives a measure of how "likely" any particular value of ${\displaystyle \,\theta \,}$ is, if we know that ${\displaystyle \,X\,}$ has the value ${\displaystyle \,x~}$. 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.^{[lower-alpha 1]} 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.^[1]

Example

Suppose

• ${\displaystyle \,X\,}$ is the number of successes in twelve independent Bernoulli trials with probability ${\displaystyle \,\theta \,}$ of success on each trial, and
• ${\displaystyle \,Y\,}$ is the number of independent Bernoulli trials needed to get three successes, again with probability ${\displaystyle \,\theta \,}$ of success on each trial (${\displaystyle \,\theta ={\tfrac {1}{2}}\;}$ for the toss of a fair coin).

Then the observation that ${\displaystyle \,X=3\,}$ induces the likelihood function

${\displaystyle {\mathcal {L}}(\theta \mid X=3)={\binom {12}{3}}\,\theta ^{3}\,(1-\theta )^{9}=220\,\theta ^{3}\,(1-\theta )^{9}~}$,

while the observation that ${\displaystyle \,Y=12\,}$ induces the likelihood function

${\displaystyle {\mathcal {L}}(\theta \mid Y=12)={\binom {11}{2}}\,\theta ^{3}\,(1-\theta )^{9}=55\,\theta ^{3}\,(1-\theta )^{9}~}$.

The likelihood principle says that, as the data are the same in both cases, the inferences drawn about the value of ${\displaystyle \,\theta \,}$ should also be the same. In addition, all the inferential content in the data about the value of ${\displaystyle \,\theta \,}$ is contained in the two likelihoods, and is the same if they are proportional to one another. This is the case in the above example, reflecting the fact that the difference between observing ${\displaystyle \,X=3\,}$ and observing ${\displaystyle \,Y=12\,}$ lies not in the actual data collected, nor in the conduct of the experimenter, but in the two different designs of the experiment.

Specifically, in one case, the decision in advance was to try twelve times, regardless of the outcome; in the other case, the advance decision was to keep trying until three successes were observed. If you support the likelihood principle then inference about ${\displaystyle \,\theta \,}$ should be the same for both cases because the two likelihoods are proportional to each other: Except for a constant leading factor of 220 vs. 55, the two likelihood functions are the same.

This equivalence is not always the case, however. The use of frequentist methods involving p-values leads to different inferences for the two cases above,^[2] showing that the outcome of frequentist methods depends on the experimental procedure, and thus violates the likelihood principle.

The law of likelihood

A related concept is the law of likelihood, the notion that the extent to which the evidence supports one parameter value or hypothesis against another is indicated by the ratio of their likelihoods, their likelihood ratio. That is,

${\displaystyle \Lambda ={{\mathcal {L}}(a\mid X=x) \over {\mathcal {L}}(b\mid X=x)}={P(X=x\mid a) \over P(X=x\mid b)}}$

is the degree to which the observation x supports parameter value or hypothesis a against b. If this ratio is 1, the evidence is indifferent; if greater than 1, the evidence supports the value a against b; or if less, then vice versa.

In Bayesian statistics, this ratio is known as the Bayes factor, and Bayes' rule can be seen as the application of the law of likelihood to inference.

In frequentist inference, the likelihood ratio is used in the likelihood-ratio test, but other non-likelihood tests are used as well. The Neyman–Pearson lemma states the likelihood-ratio test is equally statistically powerful as the most powerful test for comparing two simple hypotheses at a given significance level, which gives a frequentist justification for the law of likelihood.

Combining the likelihood principle with the law of likelihood yields the consequence that the parameter value which maximizes the likelihood function is the value which is most strongly supported by the evidence. This is the basis for the widely used method of maximum likelihood.

History

The likelihood principle was first identified by that name in print in 1962 (Barnard et al., Birnbaum, and Savage et al.), but arguments for the same principle, unnamed, and the use of the principle in applications goes back to the works of R.A. Fisher in the 1920s. The law of likelihood was identified by that name by I. Hacking (1965). More recently the likelihood principle as a general principle of inference has been championed by A. W. F. Edwards. The likelihood principle has been applied to the philosophy of science by R. Royall.^[3]

Birnbaum (1962) initially argued that the likelihood principle follows from two more primitive and seemingly reasonable principles, the conditionality principle and the sufficiency principle:

• The conditionality principle says that if an experiment is chosen by a random process independent of the states of nature ${\displaystyle \,\theta \,}$, then only the experiment actually performed is relevant to inferences about ${\displaystyle \,\theta ~}$.
• The sufficiency principle says that if ${\displaystyle \,T(X)\,}$ is a sufficient statistic for ${\displaystyle \,\theta \,}$, and if in two experiments with data ${\displaystyle x_{1}}$ and ${\displaystyle \,x_{2}\,}$ we have ${\displaystyle \,T(x_{1})=T(x_{2})\,}$, then the evidence about ${\displaystyle \,\theta \,}$ given by the two experiments is the same.

However, upon further consideration Birnbaum rejected both his conditionality principle and the likelihood principle.^[4] The adequacy of Birnbaum's original argument has also been contested by others (see below for details).

Arguments for and against

Some widely used methods of conventional statistics, for example many significance tests, are not consistent with the likelihood principle.

Let us briefly consider some of the arguments for and against the likelihood principle.

See also

• Conditionality principle
• Likelihoodist statistics

Notes

References

External links

• Anthony W.F. Edwards. "Likelihood".
• Jeff Miller. Earliest Known Uses of Some of the Words of Mathematics (L)
• John Aldrich. Likelihood and Probability in R. A. Fisher’s Statistical Methods for Research Workers