  For a general class of priors based on random series basis expansion, we
develop the Bayes Lepski's method to estimate unknown regression function. In
this approach, the series truncation point is determined based on a stopping
rule that balances the posterior mean bias and the posterior standard
deviation. Equipped with this mechanism, we present a method to construct
adaptive Bayesian credible bands, where this statistical task is reformulated
into a problem in geometry, and the band's radius is computed based on finding
the volume of certain tubular neighborhood embedded on a unit sphere. We
consider two special cases involving B-splines and wavelets, and discuss some
interesting consequences such as the uncertainty principle and self-similarity.
Lastly, we show how to program the Bayes Lepski stopping rule on a computer,
and numerical simulations in conjunction with our theoretical investigations
concur that this is a promising Bayesian uncertainty quantification procedure.
