# Hybrid Monte Carlo

In mathematics and physics, the hybrid Monte Carlo algorithm, also known as Hamiltonian Monte Carlo, is a Markov chain Monte Carlo method for obtaining a sequence of random samples from a probability distribution for which direct sampling is difficult. This sequence can be used to approximate the distribution (i.e., to generate a histogram), or to compute an integral (such as an expected value).

It differs from the Metropolis–Hastings algorithm by reducing the correlation between successive states sampled by using a Hamiltonian evolution between states and additionally by targeting states with a higher acceptance criteria than the observed probability distribution. This causes it to converge more quickly to the absolute probability distribution. It was devised by Simon Duane, A.D. Kennedy, Brian Pendleton and Duncan Roweth in 1987.[1] It proposes a state based on an arbitrary choice function $P_c$, which dictates the probability of choosing any state $i$ and then accepts or rejects the proposed state with probability

$\min\left(1,\frac{P_s(i)P_c(i \rightarrow j)}{P_s(j)P_c(j \rightarrow i)}\right)$,

this acceptance criteria has the convenient property of maintaining detailed balance for any $P_c$.

## Notes

1. ^ Duane, Simon; A.D. Kennedy, Brian J. Pendleton, and Duncan, Roweth (3 September 1987). "Hybrid Monte Carlo". Physics Letters B 195 (2): 216–222. doi:10.1016/0370-2693(87)91197-X. Retrieved 21 June 2011.