double-counting

A reader, Cory J, sent me a question in relation to a talk of mine he once attended:

I have the vague ‘memory’ of an example that was intended to bring out a central difference between broadly Bayesian methodology and broadly classical statistics.  I had thought it involved a case in which a Bayesian would say that the data should be conditionalized on, and supports H, whereas a classical statistician effectively says that the data provides no support to H.  …We know the data, but we also know of the data that only ‘supporting’ data would be given us.  A Bayesian was then supposed to say that we should conditionalize on the data that we have, even if we know that we wouldn’t have been given contrary data had it been available.

That only “supporting” data would be presented need not be problematic in itself; it all depends on how this is interpreted.  There might be no negative results to be had (H might be true) , and thus none to “be given us”.  Your last phrase, however, does describe a pejorative case for a frequentist error statistician, in that, if “we wouldn’t have been given contrary data” to H (in the sense of data in conflict with what H asserts), even “had it been available” then the procedure had no chance of finding or reporting flaws in H.  Thus only data in accordance with H would be presented, even if H is false; so H passes a “test” with minimal stringency or severity. I discuss several examples in papers below (I think the reader had in mind Mayo and Kruse 2001). Continue reading →