Magnetic Hamiltonian Monte Carlo

Hamiltonian Monte Carlo (HMC) exploits Hamiltonian dynamics to construct efficient proposals for Markov chain Monte Carlo (MCMC). In this paper, we present a generalization of HMC which exploits \textit{non-canonical} Hamiltonian dynamics. We refer to this algorithm as magnetic HMC, since in 3 dimensions a subset of the dynamics map onto the mechanics of a charged particle coupled to a magnetic field. We establish a theoretical basis for the use of non-canonical Hamiltonian dynamics in MCMC, and construct a symplectic, leapfrog-like integrator allowing for the implementation of magnetic HMC. Finally, we exhibit several examples where these non-canonical dynamics can lead to improved mixing of magnetic HMC relative to ordinary HMC.

[1]  V. Arnold.  Mathematical Methods of Classical Mechanics , 1974 .

[2]  Jiqiang Guo,et al.  Stan: A Probabilistic Programming Language. , 2017, Journal of statistical software.

[3]  Radford M. Neal.  MCMC Using Hamiltonian Dynamics , 2011, 1206.1901.

[4]  Ben Calderhead,et al.  Riemannian Manifold Hamiltonian Monte Carlo , 2009, 0907.1100.

[5]  Jascha Sohl-Dickstein,et al.  Hamiltonian Monte Carlo Without Detailed Balance , 2014, ICML.

[6]  Fang Chen,et al.  Lifting Markov chains to speed up mixing , 1999, STOC '99.

[7]  S. Duane,et al.  Hybrid Monte Carlo , 1987 .

[8]  Tianqi Chen,et al.  A Complete Recipe for Stochastic Gradient MCMC , 2015, NIPS.

[9]  K. Spiliopoulos,et al.  Irreversible Langevin samplers and variance reduction: a large deviations approach , 2014, 1404.0105.

[10]  Hans-Peter Kriegel,et al.  Integrating structured biological data by Kernel Maximum Mean Discrepancy , 2006, ISMB.

[11]  M. Girolami,et al.  Riemann manifold Langevin and Hamiltonian Monte Carlo methods , 2011, Journal of the Royal Statistical Society: Series B (Statistical Methodology).

[12]  Heikki Haario,et al.  Adaptive proposal distribution for random walk Metropolis algorithm , 1999, Comput. Stat..

[13]  Michael Betancourt,et al.  A Conceptual Introduction to Hamiltonian Monte Carlo , 2017, 1701.02434.

[14]  Jiguo Cao,et al.  Parameter estimation for differential equations: a generalized smoothing approach , 2007 .

[15]  Yichuan Zhang,et al.  Semi-Separable Hamiltonian Monte Carlo for Inference in Bayesian Hierarchical Models , 2014, NIPS.

[16]  E. Hairer,et al.  Geometric Numerical Integration , 2022, Oberwolfach Reports.

[17]  Andrew Gelman,et al.  The No-U-turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo , 2011, J. Mach. Learn. Res..

[18]  Geoffrey E. Hinton,et al.  Bayesian Learning for Neural Networks , 1995 .

[19]  T. Maiti,et al.  Regional fertility data analysis: A small area Bayesian approach , 2013 .

[20]  P. Green.  Reversible jump Markov chain Monte Carlo computation and Bayesian model determination , 1995 .

[21]  Andrew Gelman,et al.  Handbook of Markov Chain Monte Carlo , 2011 .