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

## 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.

## 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).

## 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.

## References

*arXiv:1905.11916 [Stat]*, May.

*arXiv:1701.02434 [Stat]*, January.

*Annalen Der Physik*, March.

*Bernoulli*23 (4A): 2257–98.

*arXiv Preprint arXiv:1509.07164*.

*Advances in Neural Information Processing Systems*.

*arXiv:1605.01559 [Math, Stat]*, May.

*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*73 (2): 123–214.

*Journal of Statistical Software*76 (1).

*Arxiv Preprint arXiv:1111.4246*.

*Proceedings of the 28th International Conference on Neural Information Processing Systems - Volume 2*, 2917–25. NIPS’15. Cambridge, MA, USA: MIT Press.

*arXiv:1708.07114 [Math, Stat]*, August.

*arXiv:2004.12550 [Stat]*, October.

*arXiv:1809.10756 [Cs, Stat]*, October.

*Machine Learning and the Physical Sciences Workshop at the 33rd Conference on Neural Information Processing Systems (NeurIPS)*, 8.

*Handbook for Markov Chain Monte Carlo*, edited by Steve Brooks, Andrew Gelman, Galin L. Jones, and Xiao-Li Meng. Boca Raton: Taylor & Francis.

*Biometrika*107 (2): 365–80.

*arXiv:1610.00781 [Math, Stat]*, October.

*WIREs Computational Statistics*10 (5): e1435.

*Proceedings of the 39th International Conference on Machine Learning*, 19205–20. PMLR.

*Proceedings of the 28th International Conference on Neural Information Processing Systems - Volume 1*, 955–63. NIPS’15. Montreal, Canada: MIT Press.

*Statistics & Probability Letters*91 (Supplement C): 14–19.

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