Hamiltonian and Langevin Monte Carlo

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