Bayesian methods conform to the Likelihood Principle, while frequentist methods do not. Thus, proofs of the Likelihood Principle such as Birnbaum’s (1962) appear to be threats to frequentist positions. Deborah Mayo has recently argued Birnbaum’s proof is in fact no threat to frequentist positions because it is not valid for a sampling theorist. I agree that Birnbaum’s proof is not valid for a sampling theorist, in the sense that replacing his premises with corresponding claims about what sampling distribution ought to be used for inference blocks the proof. However, I do not accept either that Birnbaum’s premises are false or that his proof as originally formulated is invalid. That being said, I do not think that Birnbaum’s proof shows that frequentist methods should not be used.
There are actually at least two different Likelihood Principles: [Read more…]