Nonparametric Bayesian Discrete Latent Variable Models for ...

Construction of A Process
B.4 Equalities and Limits
The most fundamental characteristic of a random process is the set of its finite-dimensional
distribution functions. Kolmogorov’s theorem on existence of a process is the condition
that needs to be satisfied for a family of distribution functions to define a process, see
for example Shiryaev (1995).
Exchangeability
The sequence of random variables {xi} n 1 is said to be exchangeable if any permutation
{xπi , i = 1, . . . , n} have the same joint probability distribution. The variables of
an infinite sequence are exchangeable if any finite subset of the sequence is exchangeable.
de Finetti’s representation theorem:
If the sequence of random variables {xi} n 1 is said to be infinitely exchangeable with
probability measure P , there exists a probability measure Q over the space of all distribution
functions on R such that the joint distribution function of x1, x2, . . . , xn has
the following form:
n
P (x1, . . . , xn) = F (xi) dQ(F ), (B.9)
where Q(F ) = limn→∞ P (Fn), and Fn is the empirical distribution function defined by
x1, . . . , xn.
B.4 Equalities and Limits
i=1
Recursive Property of the Gamma Function
The Gamma function has the following recursive property,
For integer n, Γ(n) = (n − 1)!.
The Dirichlet Integral
θ α1−1
1
Γ(n) = (n − 1)Γ(n − 1), (B.10)
. . . θ αn−1
n (1 − θ1 − · · · − θn) αn+1−1
dθ1 . . . dθn =
Poisson distribution as a limit of the beta distribution
n+1
The joint probability of sampling r of K new features for object i is
P (r | α, K) =
K
r
i=1 Γ(θi)
Γ n+1 i=1 θi
(B.11)
r
α/K
1 −
i + α/K
α/K
K−r . (B.12)
i + α/K
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