Hamiltonian Monte Carlo needs a proposal map that is reversible and volume-preserving

PHYSICS · Multi-part · Fermi University Physics · Computational Physics · Mechanics & Dynamical Systems · Monte Carlo and lattice simulation of physical systems

Error, Chance and the Machine, Ch Markov-chain Monte Carlo

Problem

Hamiltonian Monte Carlo needs a proposal map that is reversible and volume-preserving, and it uses the velocity Verlet (leapfrog) integrator held from the first computational-physics course for this reason, even though that integrator is not exact. For the harmonic potential U(x)=x2/2U(x) = x^2/2 (so the force is −x-x), one leapfrog step of size δt\delta t from (x0,p0)(x_0,p_0) is p1/2=p0−12δt x0,x1=x0+δt p1/2,p1=p1/2−12δt x1.\begin{aligned}p_{1/2} &= p_0 - \tfrac12\delta t\, x_0,\\ x_1 &= x_0 + \delta t\, p_{1/2},\\ p_1 &= p_{1/2} - \tfrac12\delta t\, x_1.\end{aligned} ((a)) Take δt=0.5\delta t = 0.5 and compute (x1,p1)(x_1,p_1) from (x0,p0)=(1,0)(x_0,p_0) = (1,0). ((b)) Flip the momentum, take one more leapfrog step from (x1,−p1)(x_1,-p_1), and flip the momentum again. Show that you return to (1,0)(1,0). ((c)) Write the leapfrog step as a 2×22\times2 matrix acting on (x,p)(x,p) for general δt\delta t, and show that its determinant is one. ((d)) Repeat parts (a) and (b) with one explicit Euler step, x′=x+δt px' = x + \delta t\,p, p′=p−δt xp' = p - \delta t\,x, in place of the leapfrog step, and find the determinant of the Euler step at δt=0.5\delta t = 0.5. Say which of the two properties the Euler map lacks.

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