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When I fitted the nonlinear regression using the openbugs, and calculated the 95% credible interval of the coefficient through the high density interval, I found that the total number of the credible interval including the true vales is higher than 95%, about 98%. What is the probably reason for this ?

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  • $\begingroup$ Are you asking about the frequentist properties of your credible region? $\endgroup$
    – innisfree
    Apr 21 '19 at 7:41
  • $\begingroup$ @innisfree I have read some books about the bayes, and it says the credible interval should match the nominal level to evaluate the method. So I tried to do this. $\endgroup$
    – yu zhang
    Apr 21 '19 at 10:14
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I have seen this sort of behaviour before when you do simulations and the assumed true value for the simulations is in the centre of reasonably informative priors (e.g. you have a regression coefficient with a N(0, 1) prior and simulate it as being 0). In contrast, if you sample simulation scenarios from your priors, you should on average get a coverage frequency that matches the nominal level. I.e. the reason is likely that you "sample" (or just fix them) parameter values from a much more narrow distribution at/ close to the centre of your prior.

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  • $\begingroup$ So you mean that the the true value is in the centre of my prior distribution? I will try some different true value. It is true that my prior distribution is narrow because I know that one of the regression coefficients is from 0 to 1, and it's impossible to be lower than 0 or higher than 1. So I restrict it's prior distribution to be uniform distribution(0,1) so that I can get a posterior distribution that is between 0 to 1. Does this influence its credible interval? And does this affect the coverage frequency? If it cannot match the nomial level, can we say the method is right? $\endgroup$
    – yu zhang
    Apr 21 '19 at 10:10
  • $\begingroup$ Yes, i would guess that a true value of 0.5 would give you above nominal coverage, and of course a true value of 1.001 should get you 0% coverage for that particular parameter. As said, simulating with parameter values sampled from the prior should give you exact coverage - that's what you can and should have. In a sense above nominal coverage is not a problem, is it? If course, at the edge of the prior you might get below nominal coverage. $\endgroup$
    – Björn
    Apr 21 '19 at 10:34
  • $\begingroup$ I always thought I need to set the true value to be fixed for the simulation. So you mean I can sample the true value from the prior each time and generate data, calculate its coverage interval?But that will be difficult to explain it because I don't know the true value. I am confused with the situation when the coefficient in the real world is restrict to a range. And you are right, I get below nominal coverage at the edge of the prior. $\endgroup$
    – yu zhang
    Apr 21 '19 at 15:03
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There is a nice posting that contains this already on Stack Exchange. Credible intervals are not confidence intervals. Read What's the difference between a confidence interval and a credible interval?.

Bayesian credible intervals can have better or worse coverage properties than confidence intervals. Good priors are equivalent to adding to the sample, which makes the intervals narrower and improves coverage.

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