# Rao-Blackwellization

## Recall: Rao-Blackwell Theorem

![](/files/-Luc4j5MAmYLmHctczsH)

(Lehmann & Casella, 1998)

**Rao-Blackwellization** is that

![](/files/-Luc4j5OdILLPK3W9jfA)

(Robert & Casella, 2005)

## Example: Student's $$t$$ expectation

![](/files/-Luc4j5QAKeGRJC_U6-w)

(Robert & Casella, 2005)

Here Gamma is in (α, β) form.

Alternative way to look at such decomposition

![](/files/-Luc4j5S8KjRB3VNqxxc)

(Robert & Casella, 2010)

Compare the performance with two cases,

```julia
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")
```

![](/files/-Luc4j5UP8jd25MwIYaz)


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