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Chapter 3.2
148
for our MLMM is factorized as
p(ψ) = p(w, μ 1,...,μ K, D1,...,DK, γ b, α1,...,αR,σ 2 1,...,σ 2 R, γ ε, b, u)
��K = p(w) ×
r=1
k=1
p(μk, D −1
�
k | γb) × p(γb) ��R p(αr ) × p(σ −2
�
r | γε,r ) × p(γε,r ) ×
� N �
i=1
�
p(bi | ui, μ1,...,μK, D1,...,DK) × p(ui | w) .
Particular parts of expression (7) are:
p(w) ∼D(δ,...,δ), (8)
p(μ k, D −1
k | γ b)=p(μ k) × p(D −1
k | γ b) ∼N(ξ b,k, Cb,k) ×W(ζb, Θb),
q�
Θb = diag(γb,1,...,γb,q),
q�
k =1,...,K, (9)
p(γb)= p(γb,l) ∼ G(gb,l, hb,l), (10)
l=1
l=1
pr
pr
� �
p(αr )= p(αr,l) ∼ N (ξαr ,l, c
l=1
l=1
2 αr ,l), r =1,...,R, (11)
p(σ −2
r | γε,r ) ∼G(ζε,r /2, γ −1
ε,r /2), r =1,...,R, (12)
p(γε,r ) ∼G(gε,r ,hε,r ), (13)
where D(δ,...,δ) denotes a Dirichlet distribution with parameters δ,...,δ, W(ζ, Θ)
denotes a Wishart distribution with ζ degrees of freedom and a scale matrix Θ, and
G(g, h) is a gamma distribution with parameters g and h. The last two parts of
expression (7) correspond to mixture model (3) where additionally, component allocations
u =(u1,...,uN) ′ ,ui ∈{1,...,K} (i =1,...,N) are introduced. If we use
the following prior distributions,
p(bi | ui, μ1,...,μK, D1,...,DK) ∼ s + S N (μui , Dui ), i =1,...,N, (14)
p(ui | w) ∼ P(ui = k) =wk, k =1,...,K, i =1,...,N, (15)
prior (7) whereby the vector u is integrated out, is the same as when the terms
p(bi | ui, μ1,...,μK, D1,...,DK) × p(ui | w) are replaced by p(bi | w, μ1,...,μK, D1,...,DK) ∼ s + S �K k=1 wk N (μk, Dk), i.e., by normal mixture (3).
For particular choices of the fixed hyperparameters related to the normal mixture,
δ, ξb,k, Cb,k (k = 1,...,K), gb,l, hb,l (l = 1,...,q), and leading to a weakly
(7)