  We study the problem of estimating the mode and maximum of an unknown
regression function in the presence of noise. We adopt the Bayesian approach by
using tensor-product B-splines and endowing the coefficients with Gaussian
priors. In the usual fixed-in-advanced sampling plan, we establish posterior
contraction rates for mode and maximum and show that they coincide with the
minimax rates for this problem. To quantify estimation uncertainty, we
construct credible sets for these two quantities that have high coverage
probabilities with optimal sizes. If one is allowed to collect data
sequentially, we further propose a Bayesian two-stage estimation procedure,
where a second stage posterior is built based on samples collected within a
credible set constructed from a first stage posterior. Under appropriate
conditions on the radius of this credible set, we can accelerate optimal
contraction rates from the fixed-in-advanced setting to the minimax sequential
rates. A simulation experiment shows that our Bayesian two-stage procedure
outperforms single-stage procedure and also slightly improves upon a
non-Bayesian two-stage procedure.
