The name is absent

51

posterior conditional distribution,

p(σ I xⁿ, θ*, φ, K) ex p(xⁿ ∣ θ*,φ, K, σ)p(σ ∣ θ*, φ, K)

^oc P(^σ)∏i=ιft(^χi I θ_i,σ}                    (3.22)

^oc P(σ) ∏tι ΓL∈⅛ f⅛^χi I ^θt ^σ)∙

Therefore, the Gibbs sampling scheme given by (a) and (b) is extended, when neces-
sary, by adding step (c):

(c) Generate σ conditional on the imputed values of the parameters θ*, φ and K
using the expression (3.22).

The distributions given in (3.14) and (3.21) used in steps (a) and (b) are condi-
tional on {α,7}. The Gibbs sampling scheme can be extended to include {α,7} by
adding an extra step: sampling from the appropriate posterior distribution, p(a, 7 ∣
xⁿ,θ, σ) = p(0!,7 I xⁿ,θ*,φ, K,σ). This can be done (West, 1992 and Escobar and
West, 1995 ) in the following way:

Suppose that a and 7 are a priori independent with densities

p(a,7) =p(α)p(7).

Model (3.9) implies that a and 7 given the parameters Θ*,K, φ, and σ, remain
independent. Therefore, a and 7 can be considered separately. Due to the nature of
the DP (see Antoniak, 1974), only the value of K matters in the posterior distribution
of a, i.e.,

p(o I xⁿ, θ*,φ, K, σ) — p(a ∣ K).

West (1992) proposed a model augmentation with a latent Beta random variable
for the parameter a. If the prior distribution p(α) is Ga(a, ð), then the posterior
distribution p(α ∣ K) is a mixture of gamma distributions. This allows easy simulation
from p(a I K). A new step is added to the Gibbs sampling scheme,

(d) Update the total mass parameter a:

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