$6.$ Implement a Metropolis algorithm to simulate the one$-$dimensional Ising model with

d $=20$ spins at temperature T$.$ That is$,$ the target distribution is

$\backslash$pi $($x$)$ exp

$1$

T

X

d$1$

i$=1$

xixi$+1!$

$,$

where x $=($x$1,$ x$2,...,$ xd$)$ and xi $=\backslash$pm $1,$ for all i$.$ Define h$($x$)=$

Pd$1$

i$=1$ xixi$+1.$

$($a$)$ Write down the framework of your Metropolis algorithm for this problem.

$($b$)$ Let T $=1.$ Simulate $1000$ samples after $500$ burn$-$in iterations and estimate the

average of h$($x$)$: E$\backslash$pi $[$h$($x$)].$

$($c$)$ Repeat $($b$)$ for T $=0\mathrm{.}1.$ Explain the difference in the estimated averages of h$($x$)$

for different T$.$