Riemann Manifold Langevin and Hamiltonian Monte Carlo Methods

TLDR

The paper introduces Riemann manifold Metropolis‑adjusted Langevin and Hamiltonian Monte Carlo methods to overcome limitations of existing Monte Carlo algorithms for high‑dimensional, strongly correlated target densities. These methods use the Riemann geometry of the parameter space to automatically adapt proposal dynamics, eliminating costly pilot tuning and enabling efficient exploration of complex models such as logistic regression, log‑Gaussian Cox processes, stochastic volatility, and nonlinear dynamic systems. The approach yields markedly higher time‑normalized effective sample sizes and efficient sampling even in very high dimensions, outperforming alternative algorithms. MATLAB code for the methods is available at http://www.ucl.ac.uk/statistics/research/rmhmc.

Abstract

Summary The paper proposes Metropolis adjusted Langevin and Hamiltonian Monte Carlo sampling methods defined on the Riemann manifold to resolve the shortcomings of existing Monte Carlo algorithms when sampling from target densities that may be high dimensional and exhibit strong correlations. The methods provide fully automated adaptation mechanisms that circumvent the costly pilot runs that are required to tune proposal densities for Metropolis–Hastings or indeed Hamiltonian Monte Carlo and Metropolis adjusted Langevin algorithms. This allows for highly efficient sampling even in very high dimensions where different scalings may be required for the transient and stationary phases of the Markov chain. The methodology proposed exploits the Riemann geometry of the parameter space of statistical models and thus automatically adapts to the local structure when simulating paths across this manifold, providing highly efficient convergence and exploration of the target density. The performance of these Riemann manifold Monte Carlo methods is rigorously assessed by performing inference on logistic regression models, log-Gaussian Cox point processes, stochastic volatility models and Bayesian estimation of dynamic systems described by non-linear differential equations. Substantial improvements in the time-normalized effective sample size are reported when compared with alternative sampling approaches. MATLAB code that is available from http://www.ucl.ac.uk/statistics/research/rmhmc allows replication of all the results reported.