# 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?

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.

• I obtained the posterior distribution of Dirichlet form when I did the bayesian estimation of three dependent variables with the conjugate prior; the multinomial distribution. Then I found the posterior mode for the three variables separately. Now I want to find the credibility of each posterior variables. I need help to understand the procedure / formulas if any, to find the credible intervals for each posterior estimates/modes. I have seen in many reference about 90% and 95% credible intervals. What's the difference between them? Which one is better and relevant to choose? – Anagha Raveendran Dec 26 '17 at 7:13
• 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
• The formula for Bayesian credible interval to find 95% credible interval in R is found as qbeta(.025, alpha0 + x, beta0 + n - x) and qbeta(.975, alpha0 + x, beta0 + n - x). But I am confused with the notations in the formula. I have posterior parameters (alpha1+ x1), (alpha 2 + x2) and (alpha 3 + x3). Please help me to use them in the formula. How can I find credible interval for each posterior parameters from this formula? I also have posterior modes as posterior estimates obtained from my posterior parameters respectively. Do I need them to find the credible intervals? – Anagha Raveendran Jan 2 '18 at 7:12