On the Birnbaum Argument for the Strong Likelihood Principle

The goal of this article is to provide a new clarification and critique of Birnbaum’s (1962) well-known argument purporting to show that principles of sufficiency and (weak) conditionality entail the (strong) likelihood principle. The argument has long been held as a significant result for the foundations of statistics. Not only do all of the familiar frequentist sampling theory notions violate the strong likelihood principle (SLP), the Birnbaum argument purports to show that the SLP follows from principles that sampling theorists accept. We argue that Birnbaum’s premises contain an equivocation of the vital terms leading to the unsoundness of the argument. Elucidating and extending existing critiques, we attempt to unravel the logical flaws that have enabled Birnbaum’s paradox to persist for over fifty years.