Suppose F is a class of bounded err unbounded functions. We construct a Bay
esian bootstrapped U-process over F and study the limiting behavior of the
process. W:e obtain conditional central limit theorems for Bayesian bootstr
apped U-processes and Dirichlet U-processes over F, and discuss Bayesian bo
otstrap approximations for U-processes, Some problems concerning hypothesis
testing in high-dimensional spaces are solved by combining the results in
this paper with the Projection Pursuit method.