A Hamiltonian Monte Carlo method for Bayesian Inference of Supermassive Black Hole Binaries

We investigate the use of a Hamiltonian Monte Carlo to map out the posterior density function for supermassive black hole binaries. While previous Markov Chain Monte Carlo (MCMC) methods, such as Metropolis-Hastings MCMC, have been successfully employed for a number of different gravitational wave sources, these methods are essentially random walk algorithms. The Hamiltonian Monte Carlo treats the inverse likelihood surface as a "gravitational potential" and by introducing canonical positions and momenta, dynamically evolves the Markov chain by solving Hamilton's equations of motion. We present an implementation of the Hamiltonian Markov Chain that is faster, and more efficient by a factor of approximately the dimension of the parameter space, than the standard MCMC.Comment: 16 pages, 8 figure

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