I think that I miss something very basic about the posterior distributions in bayesian hypothesis testing. In the frequentist approach, we define the null hypothesis, compute the test power, define an rejection threshold, collect the data and if the p-value is below the rejection threshold, we reject the null hypothesis.
What happens in the Bayesian case? Let's say we compare the conversion rates of two versions of a web site. We define the priors, obtain the data and get two posterior distributions, one for each version of the site. For example, the distributions look like this
What now? What can we say about the two versions? Do we "reject" one curve? Is the are common to the two curves relevant to any analysis, and if it is, what does it mean?