Abstract
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 the appropriate Riemannian metric choice for different problems. A survey of applications is provided, and some open questions are discussed.
Highlights
There are three objectives to this article
The connections between some Monte Carlo methods commonly used in statistics, physics and application domains, such as econometrics, and ideas from both Riemannian and information
We provide a full derivation of the Langevin diffusion on a Riemannian manifold and offer some intuition for how to think about such a process
Summary
There are three objectives to this article. The first is to introduce geometric concepts that have recently been employed in Monte Carlo methods based on Markov chains [1] to a wider audience. The foundations of the methods are quite different, and since the focus of the article is on using geometric ideas to improve performance, we considered a detailed description of both to be unnecessary. It should be noted, that impressive empirical evidence exists for using Hamiltonian methods in some scenarios (e.g., [4]). A key challenge in the geometric approach is which manifold to choose Throughout, π(·) will refer to an n-dimensional probability distribution and π(x) its density with respect to the Lebesgue measure
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
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 callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
