Random Walk MH

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