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slides for MCMski 4 decompression seminar at CREST

Xi'an
January 23, 2014

slides for MCMski 4 decompression seminar at CREST

the name says it all!

Xi'an

January 23, 2014
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  1. normal-normal problem
    z ∼ N(0, 1)
    θ|z ∼ N(z, 1)
    corresponding Gibbs sampler
    1. θ|z ∼ N(z, 1)
    mu.new <- rnorm(1, mean=z, sd=1)
    2. z|θ ∼ N(θ/2, 1/2)
    3. or z|θ ∼ N(0, 1)
    z <- rnorm(1, mean=0, sd=1)

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  2. normal-normal problem (2)
    in a cut model, the dependence of z on θ is “cut”
    corresponding random-walk sampler
    1. z ∼ N(0, 1)
    2. θ|z, θt−1 ∼ MH(N(z, 1); θt−1)
    mu.new <- rnorm(1, mean=mu, sd=STEP)
    R <- dnorm(z, mean=mu.new, sd=1)/dnorm(z, mean=mu,
    ifelse(runif(1) < R, mu.new, mu)
    i.e.
    1. zt ∼ N(0, 1)
    2. propose η = θt−1 + εt
    3. compute ρ = ϕ(η − zt)/ϕ(θt−1 − zt)
    4. θt = ifelse(ut < ρ, η, θt−1)

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  3. normal-normal problem (2.5)
    in a cut model, the dependence of z on θ is “cut”
    corresponding independence sampler
    1. z ∼ N(0, 1)
    2. θ|z, θt−1 ∼ MH(N(z, 1); θt−1)
    mu.new <- rnorm(1, mean=0, sd=SD)
    R <- (dnorm(mu, mean=0, sd=SD)*dnorm(z, mean=mu.new
    (dnorm(mu.new, mean=0, sd=SD)*dnorm(z, mean=mu,
    ifelse(runif(1) < R, mu.new, mu)
    i.e.
    1. zt ∼ N(0, 1)
    2. propose η ∼ N(1, 1)
    3. compute ρ = ϕ(η − zt)/ϕ(θt−1 − zt) ϕ(η − 1)/ϕ(θt−1 − 1)

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  4. normal-normal problem (end)
    different stationary distributions

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  5. Poisson-binomial case
    use of
    π(φ|z)π(θ|φ, y) =
    π(φ)f (z|φ)
    m(z)
    π(θ|φ)f (y|θ, φ)
    m(y|φ)
    instead of
    π(φ|z, y)π(θ|φ, y) ∝ π(φ)f (z|φ)π(θ|φ)f (y|θ, φ)

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