I built a Bayesian A/B testing tool - i.e. one which models A and B having posteriors $Beta(\alpha_i, \beta_i)$ where $\alpha_i, \beta_i$ are updated every iteration.
After T iterations, I compare the posterior distributions of A and B and determine which one is larger, lets say, using the mean.
How can I incorporate a third (or fourth...) group?
Originally I wanted to model it as a Dirichlet, giving each group some $\alpha$ that is updated each iteration. And then after T iterations select the group with the largest $\alpha$. But I think that it's more correct to keep each group A, B, C as a separate Beta (since they are theoretically independent) and just compare the posteriors of their respective distributions. It seems naive to me, but also intuitive. Can someone who has done this please opine?