This paper presents a theoretical analysis of numerical integration based on
interpolation with a Stein kernel. In particular, the case of integrals with
respect to a posterior distribution supported on a general Riemannian manifold
is considered and the asymptotic convergence of the estimator in this context
is established. Our results are considerably stronger than those previously
reported, in that the optimal rate of convergence is established under a basic
Sobolev-type assumption on the integrand. The theoretical results are
empirically verified on $\mathbb{S}^2$.