  This article reviews and studies the properties of Bayesian quadrature
weights, which strongly affect stability and robustness of the quadrature rule.
Specifically, we investigate conditions that are needed to guarantee that the
weights are positive or to bound their magnitudes. First, it is shown that the
weights are positive in the univariate case if the design points locally
minimise the posterior integral variance and the covariance kernel is totally
positive (e.g., Gaussian and Hardy kernels). This suggests that gradient-based
optimisation of design points may be effective in constructing stable and
robust Bayesian quadrature rules. Secondly, we show that magnitudes of the
weights admit an upper bound in terms of the fill distance and separation
radius if the RKHS of the kernel is a Sobolev space (e.g., Mat\'ern kernels),
suggesting that quasi-uniform points should be used. A number of numerical
examples demonstrate that significant generalisations and improvements appear
to be possible, manifesting the need for further research.
