Hamiltonian Monte Carlo samples the target with by simulating the Hamiltonian with a…

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

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

Problem

Hamiltonian Monte Carlo samples the target

p(x)\propto e^{-U(x)}

with

U(x) = (x^2-1)^2

by simulating the Hamiltonian

H(x,p) = \tfrac12 p^2 + U(x)

with a fictitious momentum

p

of unit mass. Starting from

x_0 = 0

p_0 = 1

, one velocity-Verlet (leapfrog) step of size

\delta t = 0.2

is taken:

\begin{aligned}p_{1/2} &= p_0 - \tfrac{\delta t}{2}U'(x_0),\\ x_1 &= x_0 + \delta t\,p_{1/2},\\ p_1 &= p_{1/2} - \tfrac{\delta t}{2}U'(x_1),\end{aligned}

with

U'(x) = 4x(x^2-1)

. ((a)) Compute

x_1

and

p_1

. ((b)) Compute

\Delta H = H(x_1,p_1) - H(x_0,p_0)

. ((c)) Give the Metropolis acceptance probability

\min(1,e^{-\Delta H})

for this proposed move.

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