Likelihood principle

Remarks

This is a principle of inference, not a theorem: it is a requirement that a method of inference may satisfy or violate. Birnbaum showed that it is implied by the sufficiency principle together with the conditionality principle; this argument and its reading are still debated. Example: one experiment tosses a coin $12$ times and sees $3$ heads; another tosses until the $3$rd head, which comes at toss $12$. The likelihoods are $\binom{12}{3} p^3 (1 - p)^9$ and $\binom{11}{2} p^3 (1 - p)^9$, which are proportional, so the principle demands the same conclusion about $p$. Bayesian inference with a fixed prior obeys the principle, since the posterior depends on the data only through the likelihood. Many frequentist procedures violate it: in the example, the two p-values for the hypothesis $p = 1/2$ differ, because they depend on outcomes that could have occurred but did not.

Going deeper

For a student

The principle separates the two main schools of statistics. A Bayesian posterior is proportional to prior times likelihood, so it respects the principle automatically, and so does a comparison of likelihood values. P-values, confidence levels and unbiasedness are defined by averaging over all possible data sets of the experiment, so they depend on the sampling design and usually violate the principle. The binomial versus negative binomial example above, with a stopping rule, is the standard illustration of this conflict.