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Is there a way to get Bayes Factor for a Mann-Whitney U ?
Yes. Johnny has developed and implemented it. I'll ask him about the current status as far as the JASP implementation is concerned (the paper is at https://arxiv.org/abs/1712.06941)
If you want to conduct the two-sample test, this is readily available in JASP. I am still in the process of implementing the one-sample versions - if you want I can link you some R-code that gives you a Bayes factor and posterior distribution.
Thanks, JonnyB. We are doing a 2-sample comparison, so can use JASP.