The name is absent

47

Expression (3.10) tells us that the probability that θ_i differs from θj, for j ≠ i, is
ɑ/(ɑ + n — 1). Moreover, denoting with go the pdf corresponding to Go,

=               +                             (3.ιι)

j≠i

The previous equation, conditional independence of æɪ,... ,x_n given θ₁,... ,θ_n and
Bayes theorem imply

p(θ_i I θ~ⁱ,x₁, ...,X_n) oc p{xι,x₂, ∙ ∙ ∙ ,x_n I θ_i,θ~^l)p(θ_i I θ~ⁱ)
<×^=₁p(^χ_j∖θ_j){ag₀(θ_i) + ∑_j^δ_θ.(θ_i)}      (3.12)

OC ap(x_i I θ_i)g₀(θ_i) + ∑_j≠_ip(xi ∣ θ_j)δ_θj(θ_i).

Let p(x_i) = ʃp(x_i I θ)go(θ) dθ denote the marginal of x_i under g₀. Then
p(%i I θ_i)g₀(θ_i) = ^Р^^Хг I ^⁹°^p{x_i) = p(θ_i I x_i)p(x_i),
P∖^xi)

we get,

p{θ_i I θ~^τ,x₁,x₂,.. .,x_n) oc q_iθgio(di) + '^Jqijδθ_j(θi),            (3.13)

j≠i

where q_io = ap(x-t) is the product of a and the marginal distribution p(x_i^y) of x_i∙,
qij = p(x_i I θj)∙, and g-to(θi) oc p(x_i ∣ ft_t)g₀(6t_i), the posterior distribution of θ_i given x_i
under a Bayes model with prior ¾ ~ G₀ and sampling model p(x_l ∣ θ_i).

We observe that a random sample of size n: θγ,θ₂,... ,θ_n of a DP can be equiva-
lently represented by the triplet (K,θ*,φ) where K is the number of distinct values
among the θ{, θ* = {θ↑,θ⅛,... ,θ*_κ} are these distinct values, and φ are indicators
φ = (<^ι, φ₂,..., φ_n), with φ_i — к if θ_t — θ_k. We refer to (K, θ*, φ) as a “configura-
tion.”

A configuration (West, 1990, MacEachern, 1994) classifies the data xⁿ {aq,..., x_n}
into K different clusters with n⅛ —     ∣ φ_i — k} observations that share the com-

mon parameter θ*_k. We use Sk to denote the ⅛-th cluster of observation indices,
S_k — {i I ψi — k}. In other words, a configuration is simply a partition of {1,..., n}

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