Sample the density

PHYSICS · Subjective · 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

Sample the density

p(x) \propto e^{-x^4}

, a stiffer well than the Gaussian of Section 11.3, with a random-walk Metropolis chain, and measure the cost of three proposal widths. (a) Use

\int_0^\infty x^k e^{-x^4}\,\mathrm{d}x = \tfrac14\Gamma\!\left(\tfrac{k+1}{4}\right)

to show that

\langle x^2\rangle = \Gamma(3/4)/\Gamma(1/4) = 0.3380

and

\langle x^4\rangle = \tfrac14

, and find

\mathrm{var}(x^2)

. (b) Write the sampler. Start at

x = 0

and take

N = 2\times10^5

steps with NumPy's default generator (PCG64), seed

2

: draw xi = rng.standard_normal(N) and then u = rng.random(N). At step

i

propose

y = x + s\,\xi_i

, accept if

u_i < \exp[-(y^4 - x^4)]

, and record

x^2

after the step, the old value again if the move is rejected. Run it with

s = 0.1

1.4

and

8.0

. (c) For each run report the acceptance, the average of

x^2

, its blocked error from

100

blocks of

2000

steps,

\tau_{\rm int}

x^2

from that error and the

\mathrm{var}(x^2)

of part (a), and

N_{\rm eff}

. (d) Which width is best? Compare with the rule of Section 11.3, using the standard deviation of the target. Give the difference of each average from the exact value in blocked errors. Then run

s = 1.4

again with seeds

3

and

4

and compare the scatter of the three averages with the blocked error.

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