I am reading Kotzen's paper Selection Bias in Likelihood Arguments.
The author takes the following principle as a starting point:
I'm confused as to how to formalize this notion in terms of Bayesian statistics, since it does not take into account the prior distributions.
More precisely, we know that
P(E|H1) α L(H1)Pr(H1)
where L(H1):=P(H1|E) is the likelihood distribution, and Pr(H1) is the prior distribution of H1. Similarly,
P(E|H2) α L(H2)Pr(H2)
The claim, then, becomes that E is evidence for H1 over H2 iff L(H1)>L(H2).
I am having a hard time understanding the term "is evidence for H1 over H2", since this can be understood in two ways:
E favors H1 over H2 if L(H1)>L(H2) means that E tilts the ratio Posterior(H1)/Posterior(H2) in favor of H1. E.g. if Posterior(H1)/Posterior(H2) = 1/2 beforehand, now we may have that Posterior(H1)/Posterior(H2) = 1, and so while beforehand Posterior(H2) was more likely, now Posterior(H1) is just as likely. In this sense, we can say E favors H1 over H2.
But then the terminology "favor" seems to suggest implicitly that Posterior(H1) > Posterior(H2) when E is presented. This interpretation takes the author to be saying: E favors H1 over H2 if L(H1)>L(H2) means that Posterior(H1) > Posterior(H2) where the posterior and likelihood are calculated w.r.t E. This seems false, since it doesn't take into account the prior; that is unless we assume a flat prior.
The question I have is:
Which of the two interpretation is meant by the Likelihood Principle?
Is my formalization of the notion correct?