  The sigma-point filters, such as the UKF, which exploit numerical quadrature
to obtain an additional order of accuracy in the moment transformation step,
are popular alternatives to the ubiquitous EKF. The classical quadrature rules
used in the sigma-point filters are motivated via polynomial approximation of
the integrand, however in the applied context these assumptions cannot always
be justified. As a result, quadrature error can introduce bias into estimated
moments, for which there is no compensatory mechanism in the classical
sigma-point filters. This can lead in turn to estimates and predictions that
are poorly calibrated. In this article, we investigate the Bayes-Sard
quadrature method in the context of sigma-point filters, which enables
uncertainty due to quadrature error to be formalised within a probabilistic
model. Our first contribution is to derive the well-known classical quadratures
as special cases of the Bayes-Sard quadrature method. Then a general-purpose
moment transform is developed and utilised in the design of novel sigma-point
filters, so that uncertainty due to quadrature error is explicitly quantified.
Numerical experiments on a challenging tracking example with misspecified
initial conditions show that the additional uncertainty quantification built
into our method leads to better-calibrated state estimates with improved RMSE.
