65
(b) Given K and φ, generate a new set of parameters (β*,δ*) from the distributions
p(∕3⅛ I N, φ, K) — Ga I βk Sβ + М2, sβtβ + ^i
for к — 1,..., K,
iesk
¼sk
and
p(⅞∣N,φ,K) = Gα
s<s + Мз, 3δ^s + Σ√
ieSk iξSk
for к = 1,..., K.
(d) Update the total mass parameter a:
(1) let a denote the currently imputed parameter value. Generate η ~ Beta(a+
l,n) and,
(2) sample the new value of a from
p(a I K, η) =πηGa(a ∖aa + K,ba- log η)
+(1 - 7Γη)Gα(α ∣ aa + K - l,ftɑ - Iogr?),
where,
7Γη ɑɑ ∣- ɪ
1 - π4 n(ba - log 77)
(e) Update hyperparameters of G0- Simulate from the complete conditional posterior
distributions,
p(tβ I M β↑-, ∙ ∙ ∙, β*κ) — Ga I tβ atg + Ksp, btfj + Sβ fik i ■
⅛=ι
p(tδ I K,δ*1,.
.., δ*κ) = Ga t<5 ats + Ksδ, bts + sδ∑δ*k
fc=l
(f) Finally, update μi and the hyperparameter tμ:
p(μ∙i I N, tμ,βi, δβ — Ga{μi ∣ Nn + Na + + sμ, 1 + ∕¾ + δi + sμtμ}.
p(tμ ∖ μ1,...,μn) = Ga I tμ atμ + nsμ,
In order to avoid the chain to get “trapped” we reinitialize the configuration every
10,000 Gibbs sampling iterations with K — n getting new values for (∕3J, jɪ ),..., (∕3*, J*)
by resampling from (3.4.2) without changing the values of the remaining parameters.