Many frequentist methods have direct Bayesian analogues, and vice versa. But there are some instances where the frequentist version is extremely cumbersome if not outright impossible. Those methods, if widely applicable and useful, could be considered system sellers of the Bayesian paradigm.
Allen Downey (whose blog post gave me the idea for this question) names Thompson sampling as essentially a sequential alternative to A/B testing where exploration is costly.
Another example, although less of a method than a general feature of Bayesian posteriors, is the ease with which uncertainty of transformed parameters can be calculated, as compared to the frequentist Delta method which is more complicated and also makes more assumptions (this example is from Kery, Introduction to WinBUGS).
I'm curious whether there are any other examples of something highly practical that is much easier in Bayesian statistics than in frequentism.
