I'm writing a term paper that I hope will serve as a preliminary step toward my philosophy comp. Here's the abstract.
Frequentist methods of statistical inference violate the likelihood principle (L). However, Birnbaum  proved that (L) follows from speciﬁc versions (S) and (C ) of two principles—the suﬃciency principle and the conditionality principle, respectively—to which frequentists appear to be committed. In a recent publication , Mayo notes that Birnbaum’s proof “has generally been accepted by frequentists, likelihoodists, and Bayesians alike” (p. 307). Nevertheless, she argues that the proof is fallacious (chapter 7(III)). Mayo’s critique involves replacing Birnbaum’s (S) and (C ) with diﬀerent formulations of the principles of suﬃciency and conditionality (S’) and (C’). Mayo shows that (S’) and (C’) do not entail
(L) but gives no reason to doubt Birnbaum’s theorem that (S) and (C) entail (L). While Mayo thus fails to show that Birnbaum’s proof is fallacious, her critique does raise the important question whether (S) and (C) or (S’) and (C’) are better formulations of the principles of suﬃciency and conditionality. I canvas a few arguments that have been oﬀered on either side of this issue. On balance, these arguments appear to favor Birnbaum’s position. However, they are not suﬃciently compelling to declare the issue resolved.
I think it's a good start. For the comp, I would like to address other responses to Birnbaum's argument in addition to Mayo's and to have something more definite to say about how we should interpret the sufficiency and conditionality principles.