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?
I don't know if you actually want to calculate a credible region for all three parameters jointly (it makes things more complicated). Instead you can calculate the credible intervals for each parameter marginally, which is simple enough since the marginal of a Dirichlet is Beta (https://math.stackexchange.com/questions/543764/let-x-y-have-a-dirichlet-distribution-with-paramters-alpha-1-alpha-2-al). There isn't a difference between credible intervals and credible regions other than the phrase "credible region" sometimes being reserved for multivariate summaries.
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.