Gaussian process regression (GPR) has been utilized to provide fast and robust estimates of plasma parameter profiles and their derivatives. We present an alternative GPR technique that performs profile regression analyses based on arbitrary linear observations. This method takes into account finite spatial resolution of diagnostics by introducing a sensitivity matrix. In addition, the profiles of interest and their derivatives can be estimated in the form of a multivariate normal distribution even when only integrated quantities are observable. We show that this GPR provides meaningful measurements of the electron density profile and its derivative in a toroidal plasma by utilizing only ten line-integrated data points given that the locations of magnetic flux surfaces are known.