>Put a flat prior on theta and you end up with an 84% posterior probability that theta is greater than 0. Step back a bit, and it’s saying that you’ll offer 5-t

>Put a flat prior on theta and you end up with an 84% posterior probability that theta is greater than 0. Step back a bit, and it’s saying that you’ll offer 5-to-1 odds that theta>0 after seeing an observation that is statistically indistinguishable from noise. That can’t make sense. Go around offering 5:1 bets based on pure noise and you’ll go bankrupt real fast. If you think it's likely to be pure noise, why the hell would you put a flat prior on it? Note also that nonflat priors are implicit in significance testing - e.g. p95 significance is similar to putting a 95% prior on the null hypothesis, and p99 significance is similar to putting a 99% prior on the null hypothesis.