The name is absent

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 θ_k^,s are a random sample from the base measure Go, The
unique values θ_k^,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, n_k^l and S_k^l 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 = _sb^e∙O ÷iT⅛M>∙   <³-i⁵>

fc=l

In words, θ_i 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 n_k∖

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(θ_n+1 I Q*, κ,φ} = ʃp(0_n+ι I G) ⅛(G I Q*, K,φ) — J G(θ_n+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[θ_n+l ∈ ∙ I Q*,φ,K∖ = E{G I Q*,φ,K} = -^~-G₀(∙') + — ∑n_k⅛V. (3.16)
a + n a + n *-~^{i k}

⅛=1

Therefore, the posterior distribution of a future observation x_n+↑ given a configuration
is

1 ^κ

Pτ[x_n+∖ ∈ ∙ I θ*,φ,K] = ^-F_n+1(. I 0_n+ι) ₊ _V_nA+i(. I 0*),   (3.17)

a + n              a + n ^z--^z

∕c=l

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