  This paper proposes and studies a numerical method for approximation of
posterior expectations based on interpolation with a Stein reproducing kernel.
Finite-sample-size bounds on the approximation error are established for
posterior distributions supported on a compact Riemannian manifold, and we
relate these to a kernel Stein discrepancy (KSD). Moreover, we prove in our
setting that the KSD is equivalent to Sobolev discrepancy and, in doing so, we
completely characterise the convergence-determining properties of KSD. Our
contribution is rooted in a novel combination of Stein's method, the theory of
reproducing kernels, and existence and regularity results for partial
differential equations on a Riemannian manifold.
