## Monday, March 28, 2011

The Wikipedia description of Lindley's Paradox asserts an example of opposite hypothesis-testing results between the Frequentist approach and the Bayesian approach.

The example is one of testing a certain town for the ratio of boy-to-girl births. The thing that violently strikes me here is the choice of the Bayesian prior: P(theta = 0.5) = 0.5, i.e., the advance assumption that it's 50% likely for the ratio to be equal to 0.5 (the other 50% chance spread uniformly between all points from 0 to 1).

I mean: What? Why would I conceivably assume that? If I broadly picture real numbers as being continuous, then my instinct would be to assume that it's almost impossible for any given number to be exactly the parameter value, i.e., I'd assume P(theta = 0.5) = 0. Even if I didn't reason that way, I otherwise have copious evidence that human births aren't really 50/50, there's very clearly more boys born than girls -- so if anything I'd choose that as the most likely prior value.

Is that really how Bayesians are supposed to choose their prior? (It seems atrocious!) Or is this just a fantastically mangled example at Wikipedia?

1. It's just a contrived example to illustrate the paradox. The paradox still works if you choose to have a prior that's a narrow Gaussian at 0.5 on top of a much broader distribution (flat, or anything wide).

The point is that the paradox most often rears its head when the prior is broad with a high narrow region in addition, and a flat prior with a delta function is just the simplest in many respects.

2. That makes a little more sense, but I'm having trouble wrapping my head around the article's general-description statements "a prior distribution that favors H0 weakly" (either example seems like favoring it strongly) and "It is a result of the prior having a sharp feature at H0" (where I'd call your example seems to have a non-sharp feature).

Thanks for addressing this -- do you have a link or citation to better presentation/example?

3. There's been some recent discussion of this http://www.science20.com/quantum_diaries_survivor/jeffreyslindley_paradox-87184