Michael: Yes, but Birnbaum purports to demonstrate, not start out with assuming, x* and y* are evidentially equivalent. He claimed, at least when he began, that his demonstration was to be relevant for non-Bayesian frequentist accounts (that it followed from principles frequentists held). That is why Savage and others heralded it as a “breakthrough”. (I say “when he began”, because I suspect he (likely) saw the gap in his argument. In any event, he rejected the SLP because it did not control error probabilities. Unfortunately, he committed suicide just a few years later.)

]]>For my response to his reply, please see my blogpost:

http://errorstatistics.com/2013/08/01/blogging-flogging-the-slp-response-to-reply-xian-robert/

Mayo

Surely a sufficient justification for reporting x* when y* was observed is the fact that they are evidentially equivalent. (Evidence apparently fails to comply with the frequency principle, but does comply with the likelihood principle.)

]]>A quick reply from my own Elba, in the Dolomiti: your arguments (about the sad consequences of the SLP) are not convincing wrt the derivation of SLP=WCP+SP. If I built a procedure that reports (E_{1},x*) whenever I observe (E_{1},x*) or (E_{2},y*), this obeys the sufficiency principle; doesn’t it? (Sorry to miss your talk!)

Sampling theorists deny that the information about the stopping rule is irrelevant for the proper inference from y*. So why should she report y* as x*? You say x* is sufficient, but that is to assume the stopping rule is irrelevant, that likelihoods alone matter (thereby begging the question). Sufficiency (weak likelihood) refers to a single experiment, SLP refers to two (with different sample spaces). Now T-B is a sufficient statistic within experiment E-B—Birnbaum’s “mathematical mixture”–that is why the inference implication from E-B differs from the inference implications from E1 as well as from E2. That blocks the SLP.

You’re right that the WCP tells us to use the known experiment (E2) in reaching the parametric inference from y*. That will differ from the inference within E-B (except of course where the unconditional happens to equal the conditional and “applying” WCP doesn’t change anything).

]]>It is true, as you say, that the WCP, once applied, tells us to output the inference implication using only the observed component, it’s Birnbaumization that asks us to consider the hypothetical mixture. Without that, anything that was an SLP violation remains one and the “proof” has not taken us anywhere. WCP was introduced for mixtures where the unconditional approach differs from the conditional, as would be the case for any SLP violation. Well, I shouldn’t try to repeat the details of the paper here… ]]>