Geometric adaptive Monte Carlo in random environment

Abstract

Manifold Markov chain Monte Carlo algorithms have been introduced to sample more effectively from challenging target densities exhibiting multiple modes or strong correlations. Such algorithms exploit the local geometry of the parameter space, thus enabling chains to achieve a faster convergence rate when measured in number of steps. However, acquiring local geometric information can often increase computational complexity per step to the extent that sampling from high-dimensional targets becomes inefficient in terms of total computational time. This paper analyzes the computational complexity of manifold Langevin Monte Carlo and proposes a geometric adaptive Monte Carlo sampler aimed at balancing the benefits of exploiting local geometry with computational cost to achieve a high effective sample size for a given computational cost. The suggested sampler is a discrete-time stochastic process in random environment. The random environment allows to switch between local geometric and adaptive proposal kernels with the help of a schedule. An exponential schedule is put forward that enables more frequent use of geometric information in early transient phases of the chain, while saving computational time in late stationary phases. The average complexity can be manually set depending on the need for geometric exploitation posed by the underlying model.

Statistical Methods for Exoplanet Detection with Radial Velocities

“…Brute-force random walk MCMC requires carefully chosen proposal distributions for systems with up to four planets (Ford 2006). Modern studies typically use ensemble samplers (Nelson et al 2014, Foreman-Mackey et al 2013, adaptive Metropolis sampling (Delisle et al 2018), Hamiltonian and/or geometric MCMC samplers (e.g., Papamarkou et al 2021), or pre-marginalisation with a Laplace approximation of the evidence over the linear parameters (Price-Whelan et al 2017). In each of these methods, the orbital periods are initialised at multiple values very near the dominant signals found by the periodogram analysis.…”