The commonly used examples of Bayesian data analysis will require specific distributional assumptions. These can be relaxed somewhat. But it is possible to provide a Bayesian analysis that does not have any assumptions about distributions. These are Bayesian analogs to various nonparametric tests. The dfba package in R implements a wide range of this Bayesian analyses. The authors of the package provide 14 (!) vignettes describing how and why you would use these analyses.
Daniel H. Barch, Richard A. Chechile. DFBA: Distribution-Free Bayesian Analysis. Published 2023-12-13. Available in html format.
An earlier version of this page was published on new.pmean.com.