A random-walk Metropolis chain with a Gaussian proposal of width samples a fixed…

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

A random-walk Metropolis chain with a Gaussian proposal of width

\sigma_{\rm prop}

samples a fixed target distribution for an observable

A

. A run of

N = 1\times10^5

steps is repeated at four proposal widths, giving the acceptance rate and the windowed estimate of

\tau_{\rm int}(A)

below:

\sigma_{\rm prop}=0.5

has acceptance

0.82

and

\tau_{\rm int}(A)=12.5

;

\sigma_{\rm prop}=1.5

has acceptance

0.51

and

\tau_{\rm int}(A)=3.0

;

\sigma_{\rm prop}=2.5

has acceptance

0.28

and

\tau_{\rm int}(A)=2.6

;

\sigma_{\rm prop}=5.0

has acceptance

0.09

and

\tau_{\rm int}(A)=6.0

. ((a)) Compute

N_{\rm eff}

for

A

at each of the four proposal widths. ((b)) State which proposal width gives the most efficient sampling of

A

for this fixed run length. ((c)) Say whether the most efficient width is the one nearest 50 per cent acceptance, and explain in one sentence why acceptance rate alone does not decide the best proposal.

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