# Approximating p-values from Bayesian posterior distribution

I'm preparing a manuscript for publication in which I have fit a mixed linear model using Bayesian regression. I'm assessing whether group A is bigger than group B. In the paper, I have reported the 95% confidence intervals of the difference for the posteriors of group A and group B. Where the 95% confidence interval doesn't overlap with 0, I state that the effect is "significant".

However, my collaborators are a bit unhappy. They want to see p-values and the names of the statistical tests. My understanding is that I could simply integrate over the posterior of the difference between group A and group B to approximate a p-value. Is this called the Bayesian posterior probability? If so, what are the rules for integrating to approximate a two-tailed t-test? Are there any good citations for this (my collaborators will insist on this)?

I realize that a Bayesian expert would know what to do, but I'm new to the Bayes approach, and this is my first paper using Bayesian regression (it's been quite the learning experience).

• If you did Bayesian linear regression, why are you talking about CI and p-values? – user2974951 Aug 27 '19 at 5:59
• This might help meet the requirements of your collaborators: A nomogram for P values by Leonhard Held bmcmedresmethodol.biomedcentral.com/articles/10.1186/… – MichiganWater Aug 27 '19 at 16:03
• @user2974951 - As pointed out in my question, my colleagues are asking for CI and p-values. While I understand that these don't really make sense in the context of Bayesian regression, I have little choice but to find a way to report these or redo my analysis using frequentist methods. – Brad Aug 27 '19 at 16:19
• I'm also a bit surprised that you feel I shouldn't be talking about CI. Doesn't it make sense to report the minimum width Bayesian credible interval along with the mean? Is it because I call it confidence interval in the question? I realize that it's not the correct term but, again, my field is extremely used to frequentist approaches so I'm trying to couch the language in terms they are more comfortable with. – Brad Aug 27 '19 at 16:22
• @user2974951 - Thank you. I talked to a statistician who suggested that I provide integrals over the posterior to provide probabilities with the understanding that most scientists in my field misinterpret p-values and what they are looking for is really the probability that the effect size falls inside a particular range. So, after talking to him I am going to write a paragraph that educates the readers (as well as my colleagues) how to interpret the 95% credible interval (which is derived from a highest posterior density estimate). – Brad Aug 28 '19 at 18:21