We study a novel 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 and variance 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, including local or piecewise polynomial regression, Lagrange interpolation, inverse distance weighting, and nearest neighbor estimation, and thus also provides a new way to estimate confidence bounds for the latter. We demonstrate the performance of MOTABAR using the reconstruction of the Lorenz attractor from noisy observations as an example.