The name is absent

49

where θ_n+ι is a new sample from G₀ and Λι+ι ɪs the cdf of x∏+ι∙ The Gibbs sampling
scheme to simulate parameters from the posterior distribution is based on the above
discussion and described next.

Gibbs Sampling Scheme: MacEachern (1994) introduced the following Gibbs sam-
pling scheme to simulate random samples θ₁,... ,θ_n from the posterior under model
(3.9). The posterior is easiest described in terms of the configuration parameters θ*,
K and φ. We are still assuming that σ, 7 and a are known, the simulation of these
parameters can be added, as we will see later, straightforward to the Gibbs sampling
scheme, so that, we keep these variables out of the notation. Call (3.13),

κ~ⁱ

p(θ_i I xⁿ, θ~ⁱ, φ~ⁱ, K~ⁱ) = q_i0g_i0(θ_i) + ∑ q_ikδ_θ→(θ_i).           (3.18)

fc=l

Under model (3.9) the weights д_гк are given by,

{cah_i(x_i), к = 0,

(3.19)
cn_k^lf_i(x_i I ¾), 1 ≤ к ≤ K~ⁱ.

Here fi is the pdf corresponding to Fi, g_lo is the posterior pdf of θ_l, obtained by
updating g₀ with the likelihood f_i(x_i ∣ θ_i), that is,

gιθ(θ_i) oc f_i(x_i I θ_i)go(θi),                             (3.20)

whose normalization constant, h_i(xi), is the marginal density of Xi,

h_i(x_i) = J f_i(x_i I θ_i)g₀(θ_i) dθ_i,

and c is a normalization constant (across к = 0,..., K~^l).

Equation (3.18) implies the posterior distribution for the indicator variables in
the configuration,

Pr{φ_i = k∖xⁿ, θ~ⁱ, φ~ⁱ, K~ⁱ) = q_ik. .           (3.21)

We can simulate samples of θ*, K and φ by iterating over the following transition
probabilities:

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