Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 216 / 4 / a / Solution

Hamiltonian Monte Carlo augments the position $x$ by an independent momentum $p\sim N(0,M)$ and uses the Hamiltonian functionFrom the current $x$, draw a fresh $p$, apply a fixed number of leapfrog steps to approximate Hamiltonian flow, and obtain $(x^{\prime },p^{\prime })$. The Metropolis–Hastings acceptance probability isotherwise retain $x$. Momentum negation may be included to make the proposal explicitly reversible. The leapfrog map is volume preserving and reversible, while the acceptance step corrects its discretization error, leaving $\pi$ invariant after the momentum is discarded.