The notation needed to explain Birnbaum's proof in detail outruns the capabilities of Blogger, so I wrote it up as a pdf. My conclusions are essentially unchanged from my rough sketch of the argument: Birnbaum's proof is valid, but I am suspicious that he has not formulated the principles of conditionality and sufficiency properly. If you interpret the principle of conditionality not as a statement about evidential equivalence but instead as a directive about how to analyze experimental results (which seems appropriate to me at this time), then it is incompatible with the principle of sufficiency as formulated by Birnbaum, and indeed with any principle that can do the work that the principle of sufficiency does in Birnbaum's proof. Another way to undermine Birnbaum's proof would be to insist, as Durbin (1970) does, that a conditional analysis can only condition on a variable that is part of the minimal sufficient statistic, although that move seems to me less appropriate to me at this time.

I did realize in examining Birnbaum's proof that it is only appropriate for experiments with discrete sample spaces. However, I do not think that this limitation is serious because the fact that no measurement is completely precise means that all real experiments have discrete sample spaces, the idea of a continuous sample space being only a useful idealization.

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