I normally use frequentist statistics but I now want to use Bayesian statistics as I want to carry out a two-sample (randomised control trial) test that includes prior information. I have an existing two-group sample and plan to collect a second two-group sample, which I would analyse for group differences, using the group differences from the first sample as the prior. (The reasons I want to do that are here but I don't think most of what's there is relevant for the current question.)
My problem is this. My standard go-to for Frequentist two-sample testing is bootstrapping the confidence interval on the differences for the means. I think this is usually better than a t-test because of having thrown out most (all?) of the parametric assumptions. But my searches for a Bayesian equivalent turn up nothing relevant. There are search results for e.g. non-parametric Bayesian or Bayesian bootstrap but they are about much more complex scenarios than a two-sample test.
I don't yet understand the Bayesian approach well but I don't see why it would inherently need to make assumptions about my data distributions. Are there methods that would give me a Bayes factor for my two-sample comparison without making parametric assumptions about the data distribution? If not, why not?
Some notes:
(1) The question is mainly to help me understand the theory, but I want to get the analysis done, and so I welcome recommendations for R code.
(2) A further complication is that my data is survey data with sample weights.
(3) My existing sample has n=500+500 and my second sample also would.
