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**) Infr_{E}(**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**

**Abstract**

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 E_{1} and E_{2} (both with unknown parameter θ), have different probability models f_{1}, f_{2}, then even though f_{1}(**x***; θ) = cf_{2}(**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 E_{i }produced the measurement, the assessment should be in terms of the properties of the particular E_{i}.

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