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).