Suppose we draw a sample of size n from a distribution with parameter \alpha, \beta. Further under Bayesian approach we are in the situation to obtain Bayesian credible/HPD interval for the parameters, not in the closed form (or may be) . Now we repeat the above procedure N number of times and obtain 95% Bayesian credible/HPD intervals. Whether the intervals will contain the true value of the parameters approximately 95% times or what type of interpretation can be made if the coverage percentage is very low, say 50% or in case of full 100% coverage.
A Bayesian credible interval does not give you a "coverage probability" - that is a frequentist statistic and as such is it not compatible with a Bayesian interpretation. The credible interval simply gives you an interval that has a specified credability of the parameter value. This is per se not related to any "true value" of the parameter but to what "you should think the parameter is (based on the information and ideas/model you have)". If you'd change the prior you would get a different credible interval.
The Bayesian interpretation does not need the imagination of a "true value". The analysis is about what one ought best to believe given some appropriate (=useful) model and some further information (data, for instance).
- Investigations based on the method to determine Bayesian confidence intervals: i. Frequentist validity of highest posterior density regions; ii. Frequentist validity of posterior quantiles (Bayesian intervals based on tails).
Some analyses show that – especially for odds ratios – the Bayesian intervals have better frequentist coverage properties if they are based on tails rather than highest posterior density regions.
- Investigations, which priors have the best frequentist properties: i. Implementing matching priors for frequentist inference; ii. Priors providing frequentist validity for Bayesian inference.
Some analyses show that non-informative priors (and even more, Jeffrey’s priors) seem to have better frequentist coverage properties. However, at the cost of loss of specific, possibly available prior information on the parameter under scrutiny.
Manfred, how will one study the "frequentist validity of a HPD region"? For instance, I don't see any justification to claim that 95% of 95%-HPDs should contain the "true" value.
If the interval (L,R) is a 95% HPD for the parameter µ that is determined from some data X, it means that
Pr(L
Sometimes even frequentists use Bayesian intervals and they want to make sure that they do not lose frequentist properties.
The connection between Bayesian confidence intervals and frequentist properties was a well-researched topic in the 1990s.
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