Nonparametric Bayesian Discrete Latent Variable Models for ...

B Mathematical Appendix
B.2 Parameterization of the Distributions Used
Beta(θ | α, β) :
Γ(α + β)
Γ(α)Γ(β) θα−1 (1 − θ) β−1
Binomial(y | π1, n) :
n
π
y
y n−y
1 (1 − π1)
k
Dirichlet(π1, . . . , πk | α1, . . . , αk) : Γ( k
j=1 αj)
k
j=1 Γ(αj)
Gamma(θ | α, β) :
1
j=1
π αj−1
j
Γ(α)βα θα−1 exp{−θ/β}
n
π yk
1 . . . πyk 1
Multinomial(y | π, n) :
y1 y2 . . . yk
1
Normal(x | µ, Σ) : exp
2π|Σ| − 1
2 (x − µ)T Σ −1 (x − µ)
B.3 Definitions
Lévy Processes
Poisson(k | λ) : λke−λ k!
Wishart(W | S, β) :
2 (βD/2) π D(D−1)/4
D β + 1 − i
−1 Γ( )
2
i=1
× |S| −β/2 |W | (β−D−1)/2 exp{− 1
2 tr(S−1 W )}
A distribution F is infinitely divisible if for every n ≥ 1, there exists a characteristic
function ψn such that
ψF (t) = ψn(t) n , (B.7)
that is, if F is the distribution of a sum of n i.i.d random variables.
Each infinitely divisible distribution corresponds to a process with stationary independent
increments, called a Lévy process. Lévy-Khintchine formula states that every
infinitely divisible distribution has a characteristic function ψ(t) = eitxF (dx) of the
form,
ψ(t) = exp ita − t 2 σ 2
/2 + [e itu − 1 − ith(u)] ν(du) , (B.8)
for a ∈ R, σ 2 ∈ R+ and a Lévy measure ν(du).
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A subordinator is a Lévy process with increasing sample paths.