Example
Suppose
- X is the number of successes in twelve independent Bernoulli trials with probability θ of success on each trial, and
- Y is the number of independent Bernoulli trials needed to get three successes, again with probability θ of success on each trial.
Then the observation that X = 3 induces the likelihood function
and the observation that Y = 12 induces the likelihood function
These are equivalent because each is a scalar multiple of the other. The likelihood principle therefore says the inferences drawn about the value of θ should be the same in both cases.
The difference between observing X = 3 and observing Y = 12 is only in the design of the experiment: in one case, one has decided in advance to try twelve times; in the other, to keep trying until three successes are observed. The outcome is the same in both cases.
Read more about this topic: Likelihood Principle
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