17,391 research outputs found
Information-geometric Markov Chain Monte Carlo methods using Diffusions
Recent work incorporating geometric ideas in Markov chain Monte Carlo is
reviewed in order to highlight these advances and their possible application in
a range of domains beyond Statistics. A full exposition of Markov chains and
their use in Monte Carlo simulation for Statistical inference and molecular
dynamics is provided, with particular emphasis on methods based on Langevin
diffusions. After this geometric concepts in Markov chain Monte Carlo are
introduced. A full derivation of the Langevin diffusion on a Riemannian
manifold is given, together with a discussion of appropriate Riemannian metric
choice for different problems. A survey of applications is provided, and some
open questions are discussed.Comment: 22 pages, 2 figure
Discussion of "Geodesic Monte Carlo on Embedded Manifolds"
Contributed discussion and rejoinder to "Geodesic Monte Carlo on Embedded
Manifolds" (arXiv:1301.6064)Comment: Discussion of arXiv:1301.6064. To appear in the Scandinavian Journal
of Statistics. 18 page
Curvature and Concentration of Hamiltonian Monte Carlo in High Dimensions
In this article, we analyze Hamiltonian Monte Carlo (HMC) by placing it in
the setting of Riemannian geometry using the Jacobi metric, so that each step
corresponds to a geodesic on a suitable Riemannian manifold. We then combine
the notion of curvature of a Markov chain due to Joulin and Ollivier with the
classical sectional curvature from Riemannian geometry to derive error bounds
for HMC in important cases, where we have positive curvature. These cases
include several classical distributions such as multivariate Gaussians, and
also distributions arising in the study of Bayesian image registration. The
theoretical development suggests the sectional curvature as a new diagnostic
tool for convergence for certain Markov chains.Comment: Comments welcom
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