The geometric foundations of Hamiltonian Monte Carlo

Betancourt, Michael; Byrne, Simon; Livingstone, Samuel; Girolami, Mark

doi:10.3150/16-bej810

Cited by 417 publications

(289 citation statements)

References 54 publications

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“…Another disadvantage is that HMC is developed using sophisticated mathematics and statistics (e.g. Betancourt et al 2014b), making it difficult to develop a deep understanding or intuition about their behaviour. We provide implementations of the static HMC and NUTS algorithms, written in R (R Core Team 2016), in Appendix B.…”

Section: Hmc In Practicementioning

confidence: 99%

Faster estimation of Bayesian models in ecology using Hamiltonian Monte Carlo

2016

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Summary1. Bayesian inference is a powerful tool to better understand ecological processes across varied subfields in ecology, and is often implemented in generic and flexible software packages such as the widely used BUGS family (BUGS, WinBUGS, OpenBUGS and JAGS). However, some models have prohibitively long run times when implemented in BUGS. A relatively new software platform called Stan uses Hamiltonian Monte Carlo (HMC), a family of Markov chain Monte Carlo (MCMC) algorithms which promise improved efficiency and faster inference relative to those used by BUGS. Stan is gaining traction in many fields as an alternative to BUGS, but adoption has been slow in ecology, likely due in part to the complex nature of HMC. 2. Here, we provide an intuitive illustration of the principles of HMC on a set of simple models. We then compared the relative efficiency of BUGS and Stan using population ecology models that vary in size and complexity. For hierarchical models, we also investigated the effect of an alternative parameterization of random effects, known as non-centering. 3.. For small, simple models there is little practical difference between the two platforms, but Stan outperforms BUGS as model size and complexity grows. Stan also performs well for hierarchical models, but is more sensitive to model parameterization than BUGS. Stan may also be more robust to biased inference caused by pathologies, because it produces diagnostic warnings where BUGS provides none. Disadvantages of Stan include an inability to use discrete parameters, more complex diagnostics and a greater requirement for hands-on tuning. 4. Given these results, Stan is a valuable tool for many ecologists utilizing Bayesian inference, particularly for problems where BUGS is prohibitively slow. As such, Stan can extend the boundaries of feasible models for applied problems, leading to better understanding of ecological processes. Fields that would likely benefit include estimation of individual and population growth rates, meta-analyses and cross-system comparisons and spatiotemporal models.

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Section: Hmc In Practicementioning

confidence: 99%

Faster estimation of Bayesian models in ecology using Hamiltonian Monte Carlo

2016

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“…As for the HMC, this algorithm produces an ergodic, time reversible Markov chain satisfying detailed balance and whose stationary marginal density is π(x) [41]. An interesting and rigorous discussion on the theoretical foundations of HMC kernels is presented in [49].…”

Section: B On Hamiltonian Based Mcmc Kernelmentioning

confidence: 99%

Langevin and Hamiltonian Based Sequential MCMC for Efficient Bayesian Filtering in High-Dimensional Spaces

Septier

Peters²

2016

IEEE J. Sel. Top. Signal Process.

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Nonlinear non-Gaussian state-space models arise in numerous applications in statistics and signal processing. In this context, one of the most successful and popular approximation techniques is the Sequential Monte Carlo (SMC) algorithm, also known as particle filtering. Nevertheless, this method tends to be inefficient when applied to high dimensional problems.In this paper, we focus on another class of sequential inference methods, namely the Sequential Markov Chain Monte Carlo (SMCMC) techniques, which represent a promising alternative to SMC methods. After providing a unifying framework for the class of SMCMC approaches, we propose novel efficient strategies based on the principle of Langevin diffusion andHamiltonian dynamics in order to cope with the increasing number of high-dimensional applications. Simulation results show that the proposed algorithms achieve significantly better performance compared to existing algorithms.

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“…The HMC method [14] generates samples from a probability density (with respect to an appropriate reference measure) known up to a constant factor by generating proposals using Hamiltonian mechanics, which is approximated by a reversible symplectic numerical integrator and followed by a Metropolis-Hastings step to correct for the bias introduced during the numerical approximations. The resulting time-homogeneous Markov chain is thus reversible, and allow distant proposals to be accepted with high probability, which decreases the correlations between samples (for a basic reference on HMC see [18], and for a geometric description see [4,6]). However, it is well-known that ergodic irreversible diffusions converge faster to their target distributions [15,21], and several irreversible MCMC algorithms based on Langevin dynamics have been proposed [19,20].…”

Section: Introductionmentioning

confidence: 99%

Irreversible Langevin MCMC on Lie Groups

Arnaudon

Barp

Takao

2019

Lecture Notes in Computer Science

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It is well-known that irreversible MCMC algorithms converge faster to their stationary distributions than reversible ones. Using the special geometric structure of Lie groups G and dissipation fields compatible with the symplectic structure, we construct an irreversible HMC-like MCMC algorithm on G, where we first update the momentum by solving an OU process on the corresponding Lie algebra g, and then approximate the Hamiltonian system on G × g with a reversible symplectic integrator followed by a Metropolis-Hastings correction step. In particular, when the OU process is simulated over sufficiently long times, we recover HMC as a special case. We illustrate this algorithm numerically using the example G = SO(3).

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The geometric foundations of Hamiltonian Monte Carlo

Cited by 417 publications

References 54 publications

Faster estimation of Bayesian models in ecology using Hamiltonian Monte Carlo

Faster estimation of Bayesian models in ecology using Hamiltonian Monte Carlo

Langevin and Hamiltonian Based Sequential MCMC for Efficient Bayesian Filtering in High-Dimensional Spaces

Irreversible Langevin MCMC on Lie Groups

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