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# Rao-Blackwellization

## Recall: Rao-Blackwell Theorem (Lehmann & Casella, 1998)
Rao-Blackwellization is that (Robert & Casella, 2005)

## Example: Student's $t$ 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,
using Distributions
using Plots
function rt_dickey(n, μ, ν, σ)
# sample y firstly
dist_y = InverseGamma(ν/2, ν/2)
ys = rand(dist_y, n)
# sample x
xs = σ * randn(n) .* sqrt.(ys) .+ μ
return xs, ys
end
function cmp_res(n, μ, ν, σ)
xs, ys = rt_dickey(n, μ, ν, σ)
cum_δm = cumsum( exp.(- xs .^2) )
cum_δm_star = cumsum( 1 ./ sqrt.(2 * σ^2 .* ys .+ 1) .* exp.(-μ^2 ./ (1 .+ 2*σ^2 .* ys)) )
δm = cum_δm ./ (1:n)
δm_star = cum_δm_star ./ (1:n)
p = plot(δm, label = "MC")
plot!(p, δm_star, label = "RB")
return p
end
using Random
Random.seed!(123)
p1 = cmp_res(10000, 0, 4.6, 1)
p2 = cmp_res(10000, 3, 5, 0.5)
plot(p1, p2)
savefig("two-situations.svg") 