Riemannian Manifold Hamiltonian Monte Carlo

By Mark A. Girolami, Ben Calderhead, Siu A. Chin. 2009

40 citations recorded by the source corpus.

The paper proposes a Riemannian Manifold Hamiltonian Monte Carlo sampler to resolve the shortcomings of existing Monte Carlo algorithms when sampling from target densities that may be high dimensional and exhibit strong correlations. The method provides a fully automated adaptation mechanism that circumvents the costly pilot runs required to tune proposal densities for Metropolis-Hastings or indeed Hybrid 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 proposed method exploits the Riemannian structure of the parameter space of statistical models and thus automatically adapts to the local manifold structure at each step based on the metric tensor. A semi-explicit second order symplectic integrator for non-separable Hamiltonians is derived for simulating paths across this manifold which provides highly efficient convergence and exploration of the target density. The performance of the Riemannian Manifold Hamiltonian Monte Carlo method is assessed by performing posterior inference on logistic regression models, log-Gaussian Cox point processes, stochastic volatility models, and Bayesian estimation of parameter posteriors of dynamical systems described by nonlinear differential equations. Substantial improvements in the time normalised Effective Sample Size are reported when compared to alternative sampling approaches. Matlab code at \url{http://www.dcs.gla.ac.uk/inference/rmhmc} allows replication of all results.

Subfields: Artificial Intelligence, Statistical and Nonlinear Physics, Statistics and Probability, Probabilistic & Bayesian Methods

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

In Statistical and Nonlinear Physics

In Statistics and Probability

In Probabilistic & Bayesian Methods

Citations in the corpus

Generalizing Hamiltonian Monte Carlo with Neural Networks (2017, methodology): “…al., 2014); with shadow Hamiltonians (Izaguirre & Hampton, 2004); with gradient pre-conditioning approaches similar to those used in Riemannian HMC (Girolami et al., 2009; Betancourt, 2013); with the use of alternative HMC accept-reject rules (Sohl-Dickstein et al., 2014; Berger et al., 2015);…”
Handbook of Markov Chain Monte Carlo (2011, neutral): “…the use of symplectic integration methods more sophisticated than the leapfrog method (eg, Creutz and Gocksch, 1989), including a recent proposal by Girolami et al. (2009) of a symplectic integrator for a non-separable Hamiltonian in which M in the kinetic energy of (2.6) depends on q, allowing…”
Variational bounds for mixed-data factor analysis (2010, methodology): “Additionally, the recently proposed Riemannian Manifold Hamiltonian Monte Carlo sampler [GCC09] may signiﬁcantly speed-up sampling-based approaches for mixed-data factor analysis models.”
Normalizing Flows on Riemannian Manifolds (2016, background): “The induced Riemannian metric Gφ allows for well-defined notions of curvature, volume, gradients of functions and divergence of vector fields [21, 22, 23].”
Magnetic Hamiltonian Monte Carlo (2016, methodology): “n explicit, symplectic integration scheme; indeed implicit integrators are used in Riemannian HMC to maintain symplecticity of the proposal which comes at a greater complexity and computational cost (Girolami et al., 2009). However explicit, symplectic integrators are simple, have good energy-conser”
A review of automatic differentiation and its efficient implementation (2018, background): “…Hamiltonian Monte Carlo (RHMC) is a Markov chain Monte Carlo sampler that requires the second and third order derivatives of the log posterior distribution, and can overcome severe geometrical pathologies ﬁrst-order sampling methods cannot (Girolami, Calderhead, & Chin, 2013; Betancourt, 2013).”
Edinburgh Research Explorer Continuous Relaxations for Discrete Hamiltonian Monte Carlo (unknown): “Continuous problems are appealing because the gradient is on your side: Unlike discrete probability mass functions, in the continuous setting, densities have derivatives, contours, and curvature that can be used to inform sampling algorithms [6, 16, 18, 20, 27].”
MCMC Using Hamiltonian Dynamics (2011, methodology): “al Hamiltonian (Izagguirre and Hampton, 2004), and the use of symplectic integration methods more sophisticated than the leapfrog method (eg, Creutz and Gocksch, 1989), including a recent proposal by Girolami et al. (2009) of a symplectic integrator for a non-separable Hamiltonian in which M in the ”
Optimal tuning of the hybrid Monte Carlo algorithm (2010, methodology): “omputational chemistry [15, 19, 27, 30], data assimilation [2], geophysics [19] and neural networks [21, 31]. The algorithm has also been proposed as a generic tool for Bayesian statistical inference [20, 6, 9]. HMC has been proposed as a method to improve on traditional Markov Chain Monte Carlo (MC”