In writing a new chapter on the Strong Likelihood Principle [i] the past few weeks, I noticed a passage in G. Casella and R. Berger (2002) that in turn recalled a puzzling remark noted in my Jan. 3, 2012 post. The post began:
A question arose from a Bayesian acquaintance:
“Although the Birnbaum result is of primary importance for sampling theorists, I’m still interested in it because many Bayesian statisticians think that model checking violates the (strong) likelihood principle (SLP), as if this principle is a fundamental axiom of Bayesian statistics”.
But this is puzzling for two reasons. First, if the LP does not preclude testing for assumptions (and he is right that it does not[ii]), then why not simply explain that rather than appeal to a disproof of something that actually never precluded model testing? To take the disproof of the LP as grounds to announce: “So there! Now even Bayesians are free to test their models” would seem only to ingrain the original fallacy.
You can read the rest of the original post here.
The remark in G. Casella and R. Berger seems to me equivocal on this point: Read more