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RichardKennaway comments on Don't You Care If It Works? - Part 1 - Less Wrong
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I think it's not going to work out. The expected posterior is equal to the prior, but the expected log Bayes factor will have the form p log(K1) + (1-p) log(K2), which for general p is just a mess. Only when p=1/2 does it simplify to log(K1 K2), and when p=1/2, K2=1/K1, so the whole thing is zero.
Okay, so I think I worked out where my failed intuition got it from. The Bayes facter is the ratio of posterior/prior for hypothesis a, divided by the ratio for hypothesis B. The top of that is expected to be 1 (because the expected posterior over the prior is one, factoring out the prior in each case keeps that fraction constant), and the bottom is also (same argument), but the expected ratio of two numbers expected to be one is not always one. So my brain turned "denominator and numerator one" into "ratio one".