A random-walk Metropolis chain samples the unit Gaussian density with proposal width

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

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

Problem

A random-walk Metropolis chain samples the unit Gaussian density

p(x) \propto e^{-x^2/2}

with proposal width

0.5

. It runs for

N = 2\times10^4

steps from

x = 0

(PCG64, seed 6). The values of

x

are cut into

b = 10

blocks of

L = 2000

steps, and the block means, to be used as printed, are

\begin{aligned}&0.094,\ 0.086,\ -0.021,\ -0.126,\ -0.078,\\ &-0.189,\ -0.055,\ 0.125,\ -0.093,\ 0.001.\end{aligned}

The variance of

x

in equilibrium is

\mathrm{var}(x) = 1

. ((a)) Find the mean

\bar x

and its blocked error from Eq. (11.12). ((b)) Merge neighbouring blocks in pairs, blocks 1 and 2, blocks 3 and 4, and so on, to make

b = 5

blocks of

L = 4000

steps, and find the blocked error again. The scatter of a plateau estimated from

b

blocks is about

1/\sqrt{2(b-1)}

of its height. Do the two errors agree within it? ((c)) Find the naive error

\sqrt{\mathrm{var}(x)/N}

. Then use the blocked error of part (a) and Eq. (11.14) to estimate

\tau_{\rm int}

and

N_{\rm eff}

x

. ((d)) Is the mean consistent with the exact value

0

? Are the blocks long enough for the blocked error to be trusted?

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