In Bayesian method, we can get a posterior distribution of a parameter. Now I want to do some simulation to know if the posterior distribution is the same as the true distribution. For example, mean of a normal distribution, In frequentist's world, I just need to set a fixed true value of mean and generate sample with this mean, calculate 95% confidence interval, repeat n times and then calculate the proportion of the confidence interval covering the true value. But I get confused with Bayesian method. If the true value is supposed to be a distribution in Bayes, do I need to generate mean from a true distribution and than generate sample according to this mean? So now every sample has different mean? Then I get a posterior distribution of the mean with Bayes. So is the posterior distribution same with the true distribution I set? But it seems not so according to the simulation I did. If I calculate 95% credible interval, what are supposed to be in this region?
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The Bayesian credible interval does not have nominal coverage, so it will not cover the true value 95% of the time (this is logically impossible, because of the prior). However, there is something similar as a coverage calculation: if you repeatedly draw parameter values from the prior, simulate data for these values, and fit the posterior, then the mixture of all posteriors will converge to the prior. Intuitively, you could say that this checks that the Bayesian estimation is unbiased, provide that data comes from the prior. See explanation here https://arxiv.org/abs/1804.06788
