  We present a nonparametric method for estimating the value and several
derivatives of an unknown, sufficiently smooth real-valued function of
real-valued arguments from a finite sample of points, where both the function
arguments and the corresponding values are known only up to measurement errors
having some assumed distribution and correlation structure. The method, Moving
Taylor Bayesian Regression (MOTABAR), uses Bayesian updating to find the
posterior mean of the coefficients of a Taylor polynomial of the function at a
moving position of interest. When measurement errors are neglected, MOTABAR
becomes a multivariate interpolation method. It contains several well-known
regression and interpolation methods as special or limit cases. We demonstrate
the performance of MOTABAR using the reconstruction of the Lorenz attractor
from noisy observations as an example.
