47
Expression (3.10) tells us that the probability that θi 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,... ,xn given θ1,... ,θn and
Bayes theorem imply
p(θi I θ~i,x1, ...,Xn) oc p{xι,x2, ∙ ∙ ∙ ,xn I θi,θ~l)p(θi I θ~i)
<×^=1p(χj∖θj){ag0(θi) + ∑j^δθ.(θi)} (3.12)
OC ap(xi I θi)g0(θi) + ∑j≠ip(xi ∣ θj)δθj(θi).
Let p(xi) = ʃp(xi I θ)go(θ) dθ denote the marginal of xi under g0. Then
p(%i I θi)g0(θi) = Р^Хг I ^9°^p{xi) = p(θi I xi)p(xi),
P∖xi)
we get,
p{θi I θ~τ,x1,x2,.. .,xn) oc qiθgio(di) + '^Jqijδθj(θi), (3.13)
j≠i
where qio = ap(x-t) is the product of a and the marginal distribution p(xiy) of xi∙,
qij = p(xi I θj)∙, and g-to(θi) oc p(xi ∣ ftt)g0(6ti), the posterior distribution of θi given xi
under a Bayes model with prior ¾ ~ G0 and sampling model p(xl ∣ θi).
We observe that a random sample of size n: θγ,θ2,... ,θn 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,..., φn), with φi — к if θt — θk. We refer to (K, θ*, φ) as a “configura-
tion.”
A configuration (West, 1990, MacEachern, 1994) classifies the data xn {aq,..., xn}
into K different clusters with n⅛ — ∣ φi — k} observations that share the com-
mon parameter θ*k. We use Sk to denote the ⅛-th cluster of observation indices,
Sk — {i I ψi — k}. In other words, a configuration is simply a partition of {1,..., n}