Rao-Blackwellization

Recall: Rao-Blackwell Theorem
(Lehmann & Casella, 1998)
Rao-Blackwellization is that
(Robert & Casella, 2005)
Example: Student's ttt
 expectation
(Robert & Casella, 2005)
Here Gamma is in (α, β) form.
Alternative way to look at such decomposition
(Robert & Casella, 2010)
Compare the performance with two cases,
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using Distributions
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using Plots
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​
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function rt_dickey(n, μ, ν, σ)
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    # sample y firstly
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    dist_y = InverseGamma(ν/2, ν/2)
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    ys = rand(dist_y, n)
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    # sample x
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    xs = σ * randn(n) .* sqrt.(ys) .+ μ
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    return xs, ys
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end
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​
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function cmp_res(n, μ, ν, σ)
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    xs, ys = rt_dickey(n, μ, ν, σ)
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    cum_δm = cumsum( exp.(- xs .^2) )
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    cum_δm_star = cumsum( 1 ./ sqrt.(2 * σ^2 .* ys .+ 1) .* exp.(-μ^2 ./ (1 .+ 2*σ^2 .* ys)) )
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    δm = cum_δm ./ (1:n)
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    δm_star = cum_δm_star ./ (1:n)
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    p = plot(δm, label = "MC")
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    plot!(p, δm_star, label = "RB")
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    return p
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end
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​
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using Random
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Random.seed!(123)
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p1 = cmp_res(10000, 0, 4.6, 1)
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p2 = cmp_res(10000, 3, 5, 0.5)
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plot(p1, p2)
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savefig("two-situations.svg")
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MC Integration

MC Optimization

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Contents

Recall: Rao-Blackwell Theorem

Example: Student's expectation

Recall: Rao-Blackwell Theorem

Example: Student's ttt expectation

Example: Student's
$t$
expectation