the likelihood principle (sequel)

As mentioned in my review of Paradoxes in Scientific Inference I was a bit confused by this presentation of the likelihood principle and this led me to ponder for a week or so whether or not there was an issue with Birnbaum’s proof (or, much more likely, with my vision of it!). After reading again Birnbaum’s proof, while sitting down in a quiet room at ICERM for a little while, I do not see any reason to doubt it. (Keep reading at your own risk!)

My confusion was caused by mixing sufficiency in the sense of Birnbaum’s mixed experiment with sufficiency in the sense of our ABC model choice PNAS paper, namely that sufficient statistics are not always sufficient to select the right model. The sufficient statistics in the proof reduces the (2,x₂) observation from Model 2 to (1,x₁) from Model 1 when there is an observation x₁ that produces a likelihood proportional to the likelihood for x₂ and the statistic is indeed sufficient: the distribution of (2,x₂) given (1,x₁) does not depend on the parameter θ. Of course, the statistic is not sufficient (most of the time) for deciding between Model 1 and Model 2, but this model choice issue is foreign to Birnbaum’s construction.