  Computation of moments of transformed random variables is a problem appearing
in many engineering applications. The current methods for moment transformation
are mostly based on the classical quadrature rules which cannot account for the
approximation errors. Our aim is to design a method for moment transformation
for Gaussian random variables which accounts for the error in the numerically
computed mean. We employ an instance of Bayesian quadrature, called Gaussian
process quadrature (GPQ), which allows us to treat the integral itself as a
random variable, where the integral variance informs about the incurred
integration error. Experiments on the coordinate transformation and nonlinear
filtering examples show that the proposed GPQ moment transform performs better
than the classical transforms.
