Hybrid Monte Carlo, or Hamiltonian Monte Carlo - University Project

Comparing the performance of Hybrid MC, Langevin MC and a simple random walk

Metropolis-Hastings Method

Random Walk Sampling

Langevin Sampling

Langevin dynamics:

Hamiltonian Sampling

Hamilton dynamics:

Where,[2]

1. The first Hamilton equation means that the force equals the negative gradient of potential energy.
2. The second Hamilton equation means that the particles velocity equals the derivative of its kinetic energy with respect to its momentum.

Numerical Methods[1]

1. Euler's Method

2. Modified Euler's Method

3. Leapfrog Method

Comparison

Reference

[1] Neal, Radford M. "MCMC using Hamiltonian dynamics, ArXiv e-prints." arXiv preprint arXiv:1206.1901 (2012).

[2] Hamiltonian mechanics