Nonparametric Bayesian Discrete Latent Variable Models for ...

B Mathematical Appendix
Taking the limit,
K
lim
K→∞ r
= lim
K→∞
= lim
K→∞
r
α/K
1 −
i + α/K
α/K
i + α/K
K! (α/i) r
1 − α/i
(K − r)!r! (K + α/i) r
K(K − 1) . . . (K − r)
(K + α/i) r
(α/i) r
r!
= (α/i)r exp{α/i}
.
r!
K−r
K + α/i
1 − α/i
K + α/i
K
1 − α/i
K + α/i
−r
K
1 − α/i
K + α/i
−r
(B.13)
The last equality is obtained from the fact that the limit of the first and the last term
is 1 and
K 1
lim
= exp{−x}.
K→∞ 1 + x/K
(B.14)
Binomial Series
(x + y) n =
n
k=0
Series expansion for the natural logarithm
124
n
x
k
k y n−k
(B.15)
ln(1 + z) = z − 1
2 z2 + 1
3 z3 − . . . , |z| ≤ 1, andz = 1 (B.16)