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What are some Bayesian ways to see if two empirical distributions come from the same data-generating process? Are any of the Bayesian alternatives implemented in Matlab/R/Python?

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Given that the two-sample Kolmogorov-Smirnov test is a nonparametric procedure, you have to dig into the area of Bayesian nonparametrics to find Bayesian analogous tools.

Here is a paper where you can start:

http://arxiv.org/abs/0910.5060

I warn you that nonparametric Bayesian theory is not easy to digest.

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