In my latest formulation of the controversial Birnbaum argument for the strong likelihood principle (SLP), I introduce a new symbol to represent a function from a given experiment-outcome pair, (E,z) to a generic inference implication. This should clarify my argument (see my new paper).
(E,z) InfrE(z) is to be read “the inference implication from outcome z in experiment E” (according to whatever inference type/school is being discussed).
A draft of my slides for the Joint Statistical Meetings JSM in Montreal next week are right after the abstract. Comments are very welcome.
Interested readers may search this blog for quite a lot of discussion of the SLP (e.g., here and here) including links to the central papers, “U-Phils” by others (e.g., here, here, and here), and amusing notes (e.g., Don’t Birnbaumize that experiment my friend, and Midnight with Birnbaum).
On the Birnbaum Argument for the Strong Likelihood Principle
An essential component of inference based on familiar frequentist notions p-values, significance and confidence levels, is the relevant sampling distribution (hence the term sampling theory). This feature results in violations of a principle known as the strong likelihood principle (SLP), the focus of this paper. In particular, if outcomes x* and y* from experiments E1 and E2 (both with unknown parameter θ), have different probability models f1, f2, then even though f1(x*; θ) = cf2(y*; θ) for all θ, outcomes x* and y* may have different implications for an inference about θ. Although such violations stem from considering outcomes other than the one observed, we argue, this does not require us to consider experiments other than the one performed to produce the data. David Cox (1958) proposes the Weak Conditionality Principle (WCP) to justify restricting the space of relevant repetitions. The WCP says that once it is known which Ei produced the measurement, the assessment should be in terms of the properties of the particular Ei.
The surprising upshot of Allan Birnbaum’s (1962) argument is that the SLP appears to follow from applying the WCP in the case of mixtures, and so uncontroversial a principle as sufficiency (SP). But this would preclude the use of sampling distributions. The goal of this article is to provide a new clarification and critique of Birnbaum’s argument. Although his argument purports that [(WCP and SP) entails SLP], we show how data may violate the SLP while holding both the WCP and SP. Such cases directly refute [WCP entails SLP].
Comments, questions, errors are welcome.
Full paper can be found here: http://arxiv-web3.library.cornell.edu/abs/1302.7021
This is so interesting. Thank you.
Fran: You’re very welcome!
Incidentally, this reminds me a story that I read about Paul Erdös and The Monty Hall problem. This is the problem for reference:
Paul Erdös got it wrong at first and he allegedly said that it should not make any difference to switch the box. I have the feeling that the reason behind Paul Erdös being unable to understand the Monty Hall problem the first time he was told is the similar to the reason behind people being unable to see why the SLP is wrong…
That is, they focus on the given data and not in how that data came to be given, and just like most people only see two doors in the Monty Hall problem and think p=1/2, those supporting SLP only see x and not how that x came to be.