Deborah Mayo’s talk in Montréal (JSM 2013)
As posted on her blog, Deborah Mayo is giving a lecture at JSM 2013 in Montréal about why Birnbaum’s derivation of the Strong Likelihood Principle (SLP) is wrong. Or, more accurately, why “WCP entails SLP”. It would have been a great opportunity to hear Deborah presenting her case and I am sorry I am missing this opportunity. (Although not sorry to be in the beautiful Dolomites at that time.) Here are the slides:
Deborah’s argument is the same as previously: there is no reason for the inference in the mixed (or Birnbaumized) experiment to be equal to the inference in the conditional experiment. As previously, I do not get it: the weak conditionality principle (WCP) implies that inference from the mixture output, once we know which component is used (hence rejecting the “and we don’t know which” on slide 8), should only be dependent on that component. I also fail to understand why either WCP or the Birnbaum experiment refers to a mixture (sl.13) in that the index of the experiment is assumed to be known, contrary to mixtures. Thus (still referring at slide 13), the presentation of Birnbaum’s experiment is erroneous. It is indeed impossible to force the outcome of y* if tail and of x* if head but it is possible to choose the experiment index at random, 1 versus 2, and then, if y* is observed, to report (E1,x*) as a sufficient statistic. (Incidentally, there is a typo on slide 15, it should be “likewise for x*”.)
Xi'an's Og 
July 31, 2013 at 5:25 pm
Christian: Something you say suggests a possible way to get to the bottom of things. You say “it is possible to choose the experiment index at random, 1 versus 2, and then, if y* is observed, to report (E1,x*) as a sufficient statistic”. I grant that the “performance” of Birnbaumization is doable in my paper. The question remains why, in interpreting the inference implication from y*, y* ought to be reported as x* and the unconditional inference computed. Imagine y* was the statistically significant outcome from the optional stopping experiment. To report it as x* is to report it came from the fixed sample size SLP pair. What justifies doing this? That is the question for you.
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).
August 1, 2013 at 9:14 pm
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 (E1,x*) whenever I observe (E1,x*) or (E2,y*), this obeys the sufficiency principle; doesn’t it? (Sorry to miss your talk!)
August 3, 2013 at 4:48 am
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
August 1, 2013 at 11:29 pm
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.)
August 3, 2013 at 4:56 am
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.)
July 31, 2013 at 4:40 am
Christian: I just came across this by chance! Quick response: I’m disappointed that you still find this unclear. The quick take is to see that counterexamples to the SLP remain, despite holding the WCP and the SP. That shows WCP + SP does not entail SLP.
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…