Hamiltonian and Langevin Monte Carlo

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.