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If frequentist confidence intervals cannot be interpreted the same as Bayesian credible intervals, why are they identical in the setting of a flat prior?
A flat prior is an attempt to express ignorance. A true informative prior is an attempt to express constraints that parameter values might take. But one Bayesian’s “informative prior” is another Bayesian’s “irrational prejudice.” There is no guarantee that Bayesians will converge when given the same information if the priors are wildly different.
Those are the philosophical arguments. In practice, Bayesian methods can work very well, and can help in evaluating frequentist procedures.
The whole point of a frequentist procedure is to avoid reliance on any prior. so the uniform prior is not practically useful. Think about Sander’s old post on credible priors:
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Sander: Then too a lot of Bayesians (e.g., Gelman) object to the uniform prior because it assigns higher prior probability to β falling outside any finite interval (−b,b) than to falling inside, no matter how large b; e.g., it appears to say that we think it more probable that OR = exp(β) > 100 or OR < 0.01 than 100>OR>0.01, which is absurd in almost every real application I’ve seen.
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Does this not suggest that values at the extremes of a frequentist confidence interval are indeed far less likely to be “true” than values closer to the point estimate? Is the point estimate therefore (assuming we truly have no prior information) not the estimate most likely to be true?
It seems almost impossible for people to resist coming to that conclusion, but it is wrong. You are not entitled to make posterior claims about the parameter given the data P(\theta | x), without making an assertion of a prior. This becomes less of a problem when large amounts of high quality data are available; they would drag any reasonable skeptical (finite) prior to the correct parameter.
If we don’t have a good idea for a prior, I prefer the Robert Matthews and I.J. Good approach, that derives skeptical and advocate priors, given a frequentist interval.
His papers recommend deriving a bound based on the result of the test; I prefer to calculate both skeptic and advocate, regardless of the reported test result.