The workshop in Chennai and its focus on sequential procedures made me realise (among other things) I had never read Cornfield’s 1966 TAS paper on sequential testing and the likelihood principle:
“By sequential analysis I mean any form of analysis in which the conclusion depends not only on the data, but also on the stopping rule.”
Written with little maths and formalism, this paper argues that keeping a fixed critical level amounts to keeping a fixed amount of evidence. Hence constituting an early critique of p-values even though not expressed in such terms. The part of the paper related with the likelihood principle does not address testing or evidence in a Bayesian way. As a side (late awakening) remark, iid observations in sequential settings are not longer independent, conditional on the stopping rule realisation N=n, since they are constrained by the fact that the stopping rule realisation is n and not n-1, n-2, … For a short while, I thought it was in turn impacting the distribution of any “sufficient” statistic one may propose, with a normalising constant that depends on the unknown parameter and hence cannot be neglected. Over all those years, I had never though of the modification of sufficiency characteristics in such contexts. But in fine the pair made of the value of the stopping rule and of the unsequential sufficient statistics proves enough. And the normalisation constant is the probability that the stopping rule.. stops!, which is equal to one! For the same short while, I was then wondering that the stopping rule principle!
“my second line of argument that there is a reasonable alternative explication of the idea of inference and one which leads to the rejection of sequential analysis. This explication is provided by the likelihood principle—which states that all observations leading to the same likelihood function should lead to the same conclusion.”
I thus went back to the fundamentals (!), namely [freely available] Bernardo’s and Smith’s Section 5.1.4 (reproduced in EJ’s Stopping rule appendix, also citing Cornfield at length), where the likelihood is properly defined by the joint density of the stopping rule τ and the attached sample at their realised values. And failing in the end (and a discussion with Judith)nto spot a missing normalisation constant.
Xi'an's Og 