Nonparametric Bayesian Discrete Latent Variable Models for ...

Using the Dirichlet integral,
= α
K
N
i=0
N
(1 − µ
i
(k))
A.2 Probability of a part of Z being inactive
α
i Γ(i + K
Γ(N + α
K
i=1
)Γ(N − i + 1)
+ 1)
= α
N N!
K (N − i)! i!
i=0
(1 − µ (k)) i (N − i)! i−1 j=0 α
K + j
N j=0 α
K + j
N!
= N j=1 α
1 +
K + j
α
N
(1 − µ
K
(k)) i
i−1 j=1 α
K + j
.
i!
Raising the above equation to the power K − k and taking the limit as K → ∞,
p(Z (:,.>k) = 0 | µ (k)) = exp − αHN + α
N
i=1
(1 − µ (k)) i
i
(A.9)
. (A.10)
The first term is obtained by using the limit given in eq. (B.14), and the second term is
obtained by the fact that the product inside the sum in the last line of eq. (A.9) reduces
to (i − 1)! in the limit.
Using the series expansion for the natural logarithm given in eq. (B.16), the second
term in the above equation can be approximated with a logarithm for large N,
p(Z (:,.>k) = 0 | µ (k)) = exp − αHN − α ln(1 − (1 − µ (k)))
=µ −α
(k) exp
− αHN .
(A.11)
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