Posterior Integration on a Riemannian Manifold

TL;DR

This paper extends an efficient posterior integration method to Riemannian manifolds, improving convergence rates for integrals on manifolds without boundary, and demonstrates its effectiveness through simulations and a paleomagnetic application.

Contribution

It generalizes a posterior integration technique from Euclidean spaces to Riemannian manifolds, eliminating boundary condition requirements for closed manifolds.

Findings

Method achieves improved convergence for integrals on manifolds.

Successfully applied to paleomagnetic data with parameters on a product manifold.

Demonstrates effectiveness through simulation studies.

Abstract

The geodesic Markov chain Monte Carlo method and its variants enable computation of integrals with respect to a posterior supported on a manifold. However, for regular integrals, the convergence rate of the ergodic average will be sub-optimal. To fill this gap, this paper extends the efficient posterior integration method of Oates et al. (2017) to the case of a Riemannian manifold. In contrast to the original Euclidean case, no non-trivial boundary conditions are needed for a closed manifold. The method is assessed through simulation and deployed to compute posterior integrals for an Australian Mesozoic paleomagnetic pole model, whose parameters are constrained to lie on the manifold $M = \mathbb{S}^2 \times \mathbb{R}_+$.