Editing Likelihood principle (section)

==Example==

Suppose

* <math>\ X\ </math> is the number of successes in twelve [[statistical independence|independent]] [[Bernoulli trial]]s with each attempt having probability <math>\ \theta\ </math> of success on each trial, and
* <math>\ Y\ </math> is the number of independent Bernoulli trials needed to get a total of three successes, again each attempt with probability <math>\ \theta\ </math> of success on each trial (if it was a fair coin each toss would have <math>\ \theta = \tfrac{\!\ 1\!\ }{ 2 }\ </math> of either outcome, heads or tails).

Then the observation that <math>\ X = 3\ </math> induces the likelihood function

:<math>\ \operatorname{\mathcal L}\left(\ \theta\ \mid\ X = 3\ \right) = \binom{12}{3} ~ \theta^3\ (1 - \theta)^9 = 220\ \theta^3\ (1-\theta)^9\ ,</math>

while the observation that <math>\ Y = 12\ </math> induces the likelihood function

:<math>\ \operatorname{\mathcal L}\left(\ \theta\ \mid\ Y = 12\ \right) = \binom{11}{2} ~ \theta^3\ (1 - \theta)^9 = 55\ \theta^3\ (1 - \theta)^9 ~.</math>

The likelihood principle says that, as the data are the same in both cases, the inferences drawn about the value of <math>\ \theta\ </math> should also be the same. In addition, all the inferential content in the data about the value of <math>\ \theta\ </math> 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 <math>\ X = 3\ </math> and observing <math>\ Y = 12\ </math> lies not in the actual data collected, nor in the conduct of the experimenter, but in the  two different [[design of experiments|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 <math>\ \theta\ </math> should be the same for both cases because the two likelihoods are proportional to each other: Except for a constant leading factor of {{math|220}} vs. {{math|55}}, the two likelihood functions are the same – constant multiples of each other.

This equivalence is not always the case, however. The use of [[frequentist]] methods involving {{nobr|[[p-values|{{mvar|p}} values]]}} leads to different inferences for the two cases above,<ref name=Vidakovic>
{{cite web
 |last = Vidakovic |first = Brani
 |title = The Likelihood Principle
 |website = H. Milton Stewart School of Industrial & Systems Engineering
 |publisher = [[Georgia Tech]]
 |url = http://www2.isye.gatech.edu/~brani/isyebayes/bank/handout2.pdf
 |access-date = 21 October 2017
}}
</ref>
showing that the outcome of frequentist methods depends on the experimental procedure, and thus violates the likelihood principle.