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Monte-Carlo
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Random Walk MH
We can write the following Julia code which use uniform distribution
${\mathcal U}[-\delta,\delta]$
as
$g$
.
# random walk Metropolis-Hastings
function rmh(T, delta, f::Function, initval = 5)
x = ones(T+1)
x[1] = initval
for t = 1:T
# generate Yt
epsi = rand() * 2 * delta - delta
Yt = epsi + x[t]
# accept or not
u = rand()
r = f(Yt)/f(x[t])
if r >= 1
x[t+1] = Yt
else
if u <= r
x[t+1] = Yt
else
x[t+1] = x[t]
end
end
end
return(x)
end
Then apply this algorithm to the normal distribution:
# density function of N(0, 1) without normalization
function dnorm(x)
return(exp(-0.5 * x^2))
end
â€‹
# example
deltalist = [0.1, 0.5, 1]
N = 15000
# results of mean and variance
muvar = zeros(3, 2)
using Statistics
ps = []
for i = 1:3
x = rmh(N, deltalist[i], dnorm)
push!(ps, plot(x, legend=:none))
muvar[i, 1] = mean(x)
muvar[i, 2] = var(x)
end
# plot
plot(ps[1], ps[2], ps[3], layout=(3,1))
savefig("res_rmh.png")
We will get the table of mean and variance as showed in Table 6.3.2 of Robert and Casella (2013)â€‹
and the curve of each case: