  Motivated by the goal of improving the efficiency of small sample design, we
propose a novel Bayesian stochastic approximation method to estimate the root
of a regression function. The method features adaptive local modelling and
nonrecursive iteration. Strong consistency of the Bayes estimator is obtained.
Simulation studies show that our method is superior in finite-sample
performance to Robbins--Monro type procedures. Extensions to searching for
extrema and a version of generalized multivariate quantile are presented.
