Likelihood principle

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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.[a] 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]



  • 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. This is the case in the above example, reflecting the fact that the difference between observing and observing lies not in the actual data collected, nor in the conduct of the experimenter, but merely in the intentions described 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. The inference about should be the same, and this is reflected in the fact that 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[edit]

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,

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.





  • The conditionality principle says that if an experiment is chosen by a random process independent of the states of nature then only the experiment actually performed is relevant to inferences about
  • The sufficiency principle says that if is a sufficient statistic for and if in two experiments with data and we have then the evidence about given by the two experiments is the same.

However, the adequacy of Birnbaum's proof is contested (see below).

Arguments for and against[edit]



The original Birnbaum argument[edit]

Birnbaum's proof of the likelihood principle has been disputed by statisticians including Michael Evans[4] and philosophers of science, including Deborah Mayo.[5][6] Alexander Dawid points out fundamental differences between Mayo's and Birnbaum's definitions of the conditionality principle, arguing Birnbaum's proof cannot be so readily dismissed.[7] A new proof of the likelihood principle has been provided by Greg Gandenberger that addresses some of the counterarguments to the original proof.[8]

Experimental design arguments on the likelihood principle[edit]


Some classical significance tests are not based on the likelihood. The following are a simple and more complicated example of those, using a commonly cited example called the optional stopping problem.

Example 1 – simple version



The likelihood function is the same in both cases: It is proportional to

So according to the likelihood principle, in either case the inference should be the same.

Example 2 – a more elaborated version of the same statistics

Suppose a number of scientists are assessing the probability of a certain outcome (which we shall call 'success') in experimental trials. Conventional wisdom suggests that if there is no bias towards success or failure then the success probability would be one half. Adam, a scientist, conducted 12 trials and obtains 3 successes and 9 failures. One of those successes was the 12th and last observation. Then Adam left the lab.

Bill, a colleague in the same lab, continued Adam's work and published Adam's results, along with a significance test. He tested the null hypothesis that p, the success probability, is equal to a half, versus p < 0.5 . If we ignore the information that the third success was the 12th and last observation the probability of the observed result that out of 12 trials 3 or something fewer (i.e. more extreme) were successes, if H0 is true, is

which is 299/4096 = 7.3% . Thus the null hypothesis is not rejected at the 5% significance level if we ignore the knowledge that the third success was the 12th result.

However observe that this first calculation also includes 12 token long sequences that end in tails contrary to the problem statement!

If we redo this calculation we realize the likelihood according to the null hypothesis must be the probability of a fair coin landing 2 or fewer heads on 11 trials multiplied with the probability of the fair coin landing a head for the 12th trial:

which is 67/20481/2 = 67/4096 = 1.64% . Now the result is statistically significant at the 5% level.


which is 134/40961/2 = 1.64% . Now the result is statistically significant at the 5% level. Note that there is no contradiction between the latter two correct analyses; both computations are correct, and result in the same p-value.

To these scientists, whether a result is significant or not does not depend on the design of the experiment, but does on the likelihood (in the sense of the likelihood function) of the parameter value being 1/2 .

Summary of the illustrated issues



The voltmeter story[edit]


An engineer draws a random sample of electron tubes and measures their voltages. The measurements range from 75 to 99 Volts. A statistician computes the sample mean and a confidence interval for the true mean. Later the statistician discovers that the voltmeter reads only as far as 100 Volts, so technically, the population appears to be “censored”. If the statistician is orthodox this necessitates a new analysis.
However, the engineer says he has another meter reading to 1000 Volts, which he would have used if any voltage had been over 100. This is a relief to the statistician, because it means the population was effectively uncensored after all. But later, the statistician infers that the second meter had not been working when the measurements were taken. The engineer informs the statistician that he would not have held up the original measurements until the second meter was fixed, and the statistician informs him that new measurements are required. The engineer is astounded. “Next you'll be asking about my oscilloscope!
Throwback to Example 2 in the prior section

This story can be translated to Adam's stopping rule above, as follows: Adam stopped immediately after 3 successes, because his boss Bill had instructed him to do so. After the publication of the statistical analysis by Bill, Adam realizes that he has missed a later instruction from Bill to instead conduct 12 trials, and that Bill's paper is based on this second instruction. Adam is very glad that he got his 3 successes after exactly 12 trials, and explains to his friend Charlotte that by coincidence he executed the second instruction. Later, Adam is astonished to hear about Charlotte's letter, explaining that now the result is significant.

See also[edit]


  1. ^ Geometrically, if they occupy the same point in projective space.


  1. ^ Dodge, Y. (2003). The Oxford Dictionary of Statistical Terms. Oxford University Press. ISBN 0-19-920613-9.
  2. ^ Vidakovic, Brani. "The Likelihood Principle" (PDF). H. Milton Stewart School of Industrial & Systems Engineering. Georgia Tech. Retrieved 21 October 2017.
  3. ^ Royall, Richard (1997). Statistical Evidence: A likelihood paradigm. Boca Raton, FL: Chapman and Hall. ISBN 0-412-04411-0.
  4. ^ Evans, Michael (2013). "What does the proof of Birnbaum's theorem prove?". arXiv:1302.5468 [math.ST].
  5. ^ Mayo, D. (2010). "An error in the argument from Conditionality and Sufficiency to the Likelihood Principle". In Mayo, D.; Spanos, A. (eds.). Error and Inference: Recent exchanges on experimental reasoning, reliability and the objectivity and rationality of science (PDF). Cambridge, GB: Cambridge University Press. pp. 305–314.
  6. ^ Mayo, D. (2014). "On the Birnbaum argument for the Strong Likelihood Principle". Statistical Science. 29: 227-266 (with discussion).
  7. ^ Dawid, A.P. (2014). "Discussion of "On the Birnbaum argument for the Strong Likelihood Principle"". Statistical Science. 29 (2): 240–241. doi:10.1214/14-STS470. S2CID 55068072.
  8. ^ Gandenberger, Greg (2014). "A new proof of the likelihood principle". British Journal for the Philosophy of Science. 66 (3): 475–503. doi:10.1093/bjps/axt039.
  • Jeffreys, Harold (1961). The Theory of Probability. The Oxford University Press.
  • Savage, Leonard J.; et al. (1962). The Foundations of Statistical Inference. London, GB: Methuen.

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