# Hamiltonian and Langevin Monte Carlo

Physics might be on to something

July 31, 2018 — November 14, 2022

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

## 1 Note salad from a Betancourt seminar

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.

## 2 Discontinuous likelihood

The solution is MOAR PHYSICS; we can construct hamiltonians which sample based on reflection/refraction dynamics in the augmented state space; see Afshar and Domke (2015);Nishimura, Dunson, and Lu (2020).

## 3 Incoming

Manifold Monte Carlo.

George Ho, Understanding NUTS and HMC

In terms of reading code, I’d recommend looking through Colin Carroll’s

`minimc`

for a minimal working example of NUTS in Python, written for pedagogy rather than actual sampling. For a “real world” implementation of NUTS/HMC, I’d recommend looking through my`littlemcmc`

for a standalone version of PyMC3’s NUTS/HMC samplers.

## 4 References

*arXiv:1905.11916 [Stat]*.

*arXiv:1701.02434 [Stat]*.

*Annalen Der Physik*.

*Bernoulli*.

*arXiv Preprint arXiv:1509.07164*.

*Advances in Neural Information Processing Systems*.

*arXiv:1605.01559 [Math, Stat]*.

*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*.

*Journal of Statistical Software*.

*Arxiv Preprint arXiv:1111.4246*.

*Proceedings of the 28th International Conference on Neural Information Processing Systems - Volume 2*. NIPS’15.

*arXiv:1708.07114 [Math, Stat]*.

*arXiv:2004.12550 [Stat]*.

*Machine Learning and the Physical Sciences Workshop at the 33rd Conference on Neural Information Processing Systems (NeurIPS)*.

*Handbook for Markov Chain Monte Carlo*.

*Biometrika*.

*arXiv:1610.00781 [Math, Stat]*.

*WIREs Computational Statistics*.

*Proceedings of the 39th International Conference on Machine Learning*.

*Proceedings of the 28th International Conference on Neural Information Processing Systems - Volume 1*. NIPS’15.

*arXiv:1809.10756 [Cs, Stat]*.

*Statistics & Probability Letters*.