54
bution,
P(φi = k∖yi, μ*-i,λ*-i, φ-i, K~i) = qik, for к = 0,.. ., K~i
where,
qi0 = со. Student ( yi ∣ m1, ~ k° , 2η1 ) ,
∖ ψ1 k0 +1 /
qik = cnkiN(yi I μ*k, λ*k} for к = 1,..., K~i,
and c is a normalization constant such that qo ÷ ∙ ■ ∙ + qχ-< — 1. Whenever we
sample φi = 0, we generate a new observation (μ*, λ*) from
<7i0(∕Aλ*)= τv(μ*∣⅞⅛i,(fco + l)λ*)
× Ga (λ* ∣ τ∕1+ n√2,≠1 + - mɪ)2)
and update, accordingly, the new configuration by K = K + 1 and φi = K ÷ 1.
(b) Given K and φ, generate a new set of parameters (μk, λ⅛) for к = 1,..., K from
the distribution
P(At⅛>λfcl yn,φ,K,k0,ψl) =
∕v(> I ¾^,C⅛ + ¾μ*)
n⅛
ko + n∣c
(mɪ - yfc)2]} ,
× Ga (λfc I 7?! + «fe/2, ≠ι + i [∑ιe,¾(yi - г/t)2 +
where yk is the mean of the observations in Sk, that is yk = ∑iesk yι∣nk-
(d) Updating the total mass parameter a:
1. Let a denote the currently imputed parameter value. Generate a latent
random variable η ~Beta(α + 1, n).
2. Sample the new value of a from
ρ(a I K, η) =πηGa(a ∖ aa + K, ba — log η')
+(1 - π^)Gα(α ∣ aa + K - 1, ba - log 77),
where,
7Γ7y aɑ I ʃf 1
1 - ⅞ n(ba - log η) '