Hamiltonian and Langevin Monte Carlo

Physics might be on to something

Hamiltonians, energy conservation in sampling. Handy. Summary would be nice.

Michael Betancourt’s heuristic explanation of Hamiltonian Monte Carlo: sets of high mass, no good - we need the “typical set”, a set whose product of differential volume and density is high. Motivates Markov Chain Monte Carlo on this basis, a way of exploring typical set given points already in it, or getting closer to the typical set if starting without. How to get a central limit theorem? “Geometric” ergodicity results. Hamiltonian Monte Carlo is a procedure for generating measure-preserving floes over phase space

\[H(q,p)=-\log(\pi(p|q)\pi(q))\] So my probability density gradient influences the particle momentum. And we can use symplectic integrators to walk through trajectories (if I knew more numerical quadrature I might know more about the benefits of this) in between random momentum perturbations. Some more stuff about resampling trajectories to de-bias numerical error, which is the NUTS extension to HMC.

Langevin Monte Carlo


To file

Manifold Monte Carlo.


Betancourt, Michael. 2017. “A Conceptual Introduction to Hamiltonian Monte Carlo.” January 9, 2017. http://arxiv.org/abs/1701.02434.
———. 2018. “The Convergence of Markov Chain Monte Carlo Methods: From the Metropolis Method to Hamiltonian Monte Carlo.” Annalen Der Physik, March. https://doi.org/10.1002/andp.201700214.
Betancourt, Michael, Simon Byrne, Sam Livingstone, and Mark Girolami. 2017. “The Geometric Foundations of Hamiltonian Monte Carlo.” Bernoulli 23 (November): 2257–98. https://doi.org/10.3150/16-BEJ810.
Carpenter, Bob, Matthew D. Hoffman, Marcus Brubaker, Daniel Lee, Peter Li, and Michael Betancourt. 2015. “The Stan Math Library: Reverse-Mode Automatic Differentiation in C++.” 2015. http://arxiv.org/abs/1509.07164.
Durmus, Alain, and Eric Moulines. 2016. “High-Dimensional Bayesian Inference via the Unadjusted Langevin Algorithm.” May 5, 2016. http://arxiv.org/abs/1605.01559.
Girolami, Mark, and Ben Calderhead. 2011. “Riemann Manifold Langevin and Hamiltonian Monte Carlo Methods.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 73 (2): 123–214. https://doi.org/10.1111/j.1467-9868.2010.00765.x.
Goodrich, Ben, Andrew Gelman, Matthew D. Hoffman, Daniel Lee, Bob Carpenter, Michael Betancourt, Marcus Brubaker, Jiqiang Guo, Peter Li, and Allen Riddell. 2017. “Stan : A Probabilistic Programming Language.” Journal of Statistical Software 76 (1). https://doi.org/10.18637/jss.v076.i01.
Neal, Radford M. 2011. MCMC Using Hamiltonian Dynamics.” In Handbook for Markov Chain Monte Carlo, edited by Steve Brooks, Andrew Gelman, Galin L. Jones, and Xiao-Li Meng. Boca Raton: Taylor & Francis. http://arxiv.org/abs/1206.1901.
Norton, Richard A., and Colin Fox. 2016. “Tuning of MCMC with Langevin, Hamiltonian, and Other Stochastic Autoregressive Proposals.” October 3, 2016. http://arxiv.org/abs/1610.00781.
Xifara, T., C. Sherlock, S. Livingstone, S. Byrne, and M. Girolami. 2014. “Langevin Diffusions and the Metropolis-Adjusted Langevin Algorithm.” Statistics & Probability Letters 91 (August): 14–19. https://doi.org/10.1016/j.spl.2014.04.002.

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