“Due to the various pragmatic obstacles, it is rare for a mission-critical analysis to be done in the “fully Bayesian” manner, i.e., without the use of tried-and-true frequentist tools at the various stages. Philosophy and beauty aside, the reliability and efficiency of the underlying computations required by the Bayesian framework are the main practical issues. A central technical issue at the heart of this is that it is much easier to do optimization (reliably and efficiently) in high dimensions than it is to do integration in high dimensions. Thus the workhorse machine learning methods, while there are ongoing efforts to adapt them to Bayesian framework, are almost all rooted in frequentist methods. A work-around is to perform MAP inference, which is optimization based.
Most users of Bayesian estimation methods, in practice, are likely to use a mix of Bayesian and frequentist tools. The reverse is also true—frequentist data analysts, even if they stay formally within the frequentist framework, are often influenced by “Bayesian thinking,” referring to “priors” and “posteriors.” The most advisable position is probably to know both paradigms well, in order to make informed judgments about which tools to apply in which situations.”