The name is absent

46

p(γ). The general DPM becomes

I. x_i I θ_i, σ ~ F_i(x_i I θ_i, σ) and are independent, for i = 1,..., n,

II. 0₁,...,0_n∣G⅛^dGandσ~p(σ),

III. G∣α,7~DP(α,G0(∙,∣7))                                  '

IV. a ~ p(ct) and 7 ~ p(7).

The levels I-III (with a and 7 fixed) correspond to the model in Lo(1984), while levels
II-IV correspond, essentially, to the Antoniak (1974) model. Escobar and West (1995)
join both models including all the levels (I-IV). Fi determines if the distribution of ¾
is continuous or discrete. In the following subsection we describe the Gibbs sampling
scheme for posterior simulation of 0₁,..., θ_n.

3.2.3 Gibbs Sampling Scheme for DPM

This subsection summarizes posterior MCMC simulation in DP mixture models as
described in West et al. (1994). We describe posterior MCMC in the mixture model
defined in (3.9).

Complete conditional posterior distribution of θ_i∙. We use transition probabili-
ties defined by sampling θ_i from its complete conditional posterior given the currently
imputed values of the other 0∕s, all other parameters and the observed data. For the
moment, we assume that the parameters a and σ are fixed and suppress them in the
notation. We will include them later.

Assume θ_i ~ G and G ~ DP(a, Go). Let θ~^l = (0_x,..., θ_i~ι, θ_i+ι,..., θ_n). The
updating rule (3.4) of the DP implies, G ∣ в~^г ~ DP(α + n — 1, aGo + ∑_j≠_i <⅛)- W^e
can marginalize G, using the Polya urn, and find

≡ ∙ I «-I = ∏⅛^{w +}      ∑ ⅛<∙)∙ (ɜ'ɪo)

j≠i

That is, we can directly simulate from (¾ ∣ θ~^l, without any need to generate the
RPM G.

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