# 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.

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## To file

Manifold Monte Carlo.

## References

Betancourt, Michael. 2017. arXiv:1701.02434 [Stat], January.
———. 2018. Annalen Der Physik, March.
Betancourt, Michael, Simon Byrne, Sam Livingstone, and Mark Girolami. 2017. Bernoulli 23 (4A): 2257–98.
Carpenter, Bob, Matthew D. Hoffman, Marcus Brubaker, Daniel Lee, Peter Li, and Michael Betancourt. 2015. arXiv Preprint arXiv:1509.07164.
Durmus, Alain, and Eric Moulines. 2016. arXiv:1605.01559 [Math, Stat], May.
Girolami, Mark, and Ben Calderhead. 2011. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 73 (2): 123–214.
Goodrich, Ben, Andrew Gelman, Matthew D. Hoffman, Daniel Lee, Bob Carpenter, Michael Betancourt, Marcus Brubaker, Jiqiang Guo, Peter Li, and Allen Riddell. 2017. Journal of Statistical Software 76 (1).
Neal, Radford M. 2011. In Handbook for Markov Chain Monte Carlo, edited by Steve Brooks, Andrew Gelman, Galin L. Jones, and Xiao-Li Meng. Boca Raton: Taylor & Francis.
Norton, Richard A., and Colin Fox. 2016. arXiv:1610.00781 [Math, Stat], October.
Xifara, T., C. Sherlock, S. Livingstone, S. Byrne, and M. Girolami. 2014. Statistics & Probability Letters 91 (Supplement C): 14–19.

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