# The Bayes credible interval / Bayes credible region of the posterior distribution of a multinomial Dirichlet conjugate pairs

I have a posterior distribution of Dirichlet form with new parameters (alpha1, x1), (alpha 2, x2) and (alpha 3, x3) and the posterior mode of each dependent variable as the Bayes estimator. I wish to find the Bayes credible interval/Bayes credible region of the posterior to know the uncertainty in my Bayes estimators. But I have no idea how to find it for each of the three posterior estimators of three dependent variables I have. I am also confused with 95% and 90% credible interval. Why there is no 100% probability that a posterior estimator lies in an interval? What is the difference between Bayes credible interval and Bayes credible region?

As for why there aren't any 100% credible intervals that get reported, think about it for a second. If you'd like, for example, the 100% credible interval for one of the parameters (which is a proportion) in your problem marginally, it's just going to be $[0,1]$! If instead we wanted a 100% credible interval for some posterior that's normal, it'd just be $(-\infty,\infty)$! In general, a 100% credible interval is going to just spit back out the interval where the prior has any mass, which isn't helpful at all.
• As I said, each parameter is marginally beta (with parameters described in the link above), so all you have to do to get the (1-$\alpha$)% credible interval for each parameter is to calculate the $\alpha/2$ and $1-\alpha/2$ quantiles of the marginal beta (which can be done in one line in R). Deciding which $\alpha$ is best for you depends on your problem at hand! $95$% is somewhat standard, but it's really up to you. – aleshing Dec 26 '17 at 16:34