Confidence Bounds for Spherical SamplingSchemes in Stereology

This paper considers quadrature methods for integration over the unit sphere using reproducing kernel Hilbert spaces. Bayesian quadrature weights, which minimize the worst-case integration error, and their normalized counterparts, which additionally integrate constants exactly, are studied. Deterministic error bounds are established in terms of the Sobolev regularity of the integrand and geometric properties of the sampling design. For designs obtained by applying a uniformly random rotation to a fixed point configuration, exact mean and bias identities are derived, and the variance is bounded by the squared worst-case error. The normalized estimator is unbiased, whereas the bias of the unconstrained Bayesian estimator is proportional to its squared worst-case error. To construct data-dependent bounds, an ordinary least-squares estimator of the spherical harmonic coefficients is introduced, yielding a conservative 95\% confidence interval under an assumption on the Sobolev norm of the harmonic tail. The framework is applied to the Nucleator in stereology, for which explicit error bounds are obtained for convex bodies in dimensions two and three.