Nonparametric Bayesian Discrete Latent Variable Models for ...

A Details of Derivations for the
Stick-Breaking Representation of IBP
The stick-breaking construction of the IBP has been summarized in Section 4.1.2. Here,
we give details of derivation for obtaining this representation.
A.1 Densities of ordered stick lengths
The density for the kth largest feature presence probability is obtained by considering
the decreasing order statistics of K i.i.d. random variables in the limit as K → ∞. We
define µl1 ≥ µl2 ≥ · · · ≥ µlK to be the decreasing ordering of µ1, . . . , µK, and Lk to be
the set of indices other than the k largest µ’s,
Lk = {1, . . . , K}\{l1, . . . , lk}.
Thus, the indices for the k largest feature presence probabilities are {l1, . . . , lk}. Denoting
µ (k) = µlk , the k largest values among µ1:K is denoted as µ (1:k) = {µ (1), . . . , µ (k)}
and we have
µl ≤ µ (k) for all l ∈ Lk. (A.1)
Thus, the range of the subsequent µl’s given µ (k) is restricted to [0, µ (k)], resulting in
the following pdf:
p(µl | µ (1:k)) =
α
K
µ(k)
0
µ α
K −1
l
α α
t K
K −1 dt
= α
K
µ− α
K
(k)
Integrating, the cdf for µl conditioned on µ (1:k) is found to be
F (µl | µ (1:k)) =
= µ
µl
0
α
− K
(k)
α
K
α
K
µ−
(k)
t α
K −1 dt
µ α
K
l I(0 ≤ µl ≤ µ (k)) + I(µ (k) < µl).
µ α
K −1
l . (A.2)
(A.3)
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