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A specific method? No, obviously. That depends on the problem. What I would love to see is:
"H0, H1 and H2 are our prior hypothesis. We assume them to be mutually exclusive and complete. H0, based on these empirical assesments, has probability P0. H1, base on those other considerations, has probability P1. H2 intercepts all the other possible explanations and has probability 1 - P0 - P1.
We are going to use this method to analyze the data.
These are the data.
Based on the calculations, the revised probabilities for the three hypothesis are P0', P1' and P2'."
How about: we only know the mean of the effect, so we suppose an exponential prior distribution. Or: we value error quadratically, so we apply a normal distribution? Or: we know that the effect stays the same at every time scale, so we are going to start from a Poisson distribution?
When it comes to a test problem, what about an antidepression drug?
Who's that in case of a FDA approval process? The person who wants his drug approved or the FDA? If it's the person who wants his drug approved, why don't they just go into it with strong priors?