Abstract
Markov chain Monte Carlo methods explicitly defined on the manifold of probability distributions have recently been established. These methods are constructed from diffusions across the manifold and the solution of the equations describing geodesic flows in the Hamilton–Jacobi representation. This paper takes the differential geometric basis of Markov chain Monte Carlo further by considering methods to simulate from probability distributions that themselves are defined on a manifold, with common examples being classes of distributions describing directional statistics. Proposal mechanisms are developed based on the geodesic flows over the manifolds of support for the distributions, and illustrative examples are provided for the hypersphere and Stiefel manifold of orthonormal matrices.
Highlights
Markov chain Monte Carlo (MCMC) methods that originated in the physics literature have caused a revolution in statistical methodology over the last 20 years by providing the means, in an almost routine manner, to perform Bayesian inference over arbitrary non-conjugate prior and posterior pairs of distributions (Gilks et al, 1996)
The tangent space at x 2 Sd 1 is the .d 1/-dimensional subspace of vectors orthogonal to x: Tx D 1v 2 Rd W x>v D 0o: A Riemannian manifold incorporates a notion of distance, such that for a point q 2 M, there exists a positive-definite matrix G, called the metric tensor, that forms an inner product between tangents u and v hu; viG D u>G.q/v: Information geometry is the application of differential geometry to families of probability distributions
Riemannian manifold Hamiltonian Monte Carlo (RMHMC) is an MCMC scheme whereby new samples are proposed by approximately solving a system of differential equations describing the paths of Hamiltonian dynamics on the manifold (Girolami and Calderhead, 2011)
Summary
Markov chain Monte Carlo (MCMC) methods that originated in the physics literature have caused a revolution in statistical methodology over the last 20 years by providing the means, in an almost routine manner, to perform Bayesian inference over arbitrary non-conjugate prior and posterior pairs of distributions (Gilks et al, 1996). The HMC proposal mechanism is based on simulating Hamiltonian dynamics defined by the target distribution (see Neal (2011) for a comprehensive tutorial). For this reason, HMC is routinely referred to as Hamiltonian Monte Carlo. Scand J Statist 40 on manifolds embedded in Euclidean space, by exploiting the existence of explicit forms for geodesics. This can provide a significant boost in speed, by avoiding the need to solve large linear systems as well as complications arising because of the lack of a single global coordinate system.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
