48
into clusters defined by the unique values 0*, together with a list of these unique val-
ues. In the Gibbs sampling scheme we will update, in the first step, the configuration
given the previous one, and, in the second step, θ↑, ■ ■ ∙ > given φ and K.
Moreover, since the θk,s are a random sample from the base measure Go, The
unique values θk,s are conditionally independent given φ, with posterior densities:
Ж I xn, φ, K) ex p(xn I θ*k, φ, K)p(θ*k I φ, K)
Г 1 (3∙14)
<× {∏i∈⅜Λ(¾ I θ*k)}9o{θ*k).
Denoting by K~τ, nkl and Skl for k = 1,..., K~l and θ*~t := (θ↑~∖ ..., θ*κ-i) the
configuration corresponding to the random sample θ~^l, the conditional prior (3.10) is
equivalent to
p^‘ ≡ ■ I «“‘1 = sbe∙O ÷iT⅛M>∙ <3-i5>
fc=l
In words, θi is different from the other parameters and drawn from G0 with probability
proportional to a, and otherwise equal to the /с-th already observed value, θ*k~l, with
probability proportional to the number of times this value has been observed in the
sample 0~г, i.e., oc nk∖
The extension of the expression (3.15) from n to n + 1 yields to the predictive
distribution of a new value Qi with i — ∏+ 1. This distribution is identical to the
expected value of G given Q*,φ, K. This is easily seen by
p(θn+1 I Q*, κ,φ} = ʃp(0n+ι I G) ⅛(G I Q*, K,φ) — J G(θn+1) dp(G ∣ Q*, K, φ) = G.
Thus, once we have it, the posterior sample of the parameters can be used to estimate
G. The predictive distribution (on the random effects) is
1 κ
Pr[θn+l ∈ ∙ I Q*,φ,K∖ = E{G I Q*,φ,K} = -^~-G0(∙') + — ∑nk⅛V. (3.16)
a + n a + n *-~i k
⅛=1
Therefore, the posterior distribution of a future observation xn+↑ given a configuration
is
1 κ
Pτ[xn+∖ ∈ ∙ I θ*,φ,K] = ^-Fn+1(. I 0n+ι) + _VnA+i(. I 0*), (3.17)
a + n a + n z--z
∕c=l
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