Source
Thibaux and Jordan (2007), AISTATS, PMLR 2: 564–571, LIT-1537, read in full in NOTE-1171; equations (5)–(7). The IBP it is matched to is Griffiths and Ghahramani’s (LIT-1556, §4.4). Conjugacy (5) is cited in the paper from Kim (1999, theorem 3.3) and Hjort (1990).
What was actually shown
Take c constant and B0 continuous with mass γ. Let B ~ BP(c, B0) and X1, …, Xn | B ~ BeP(B) independently.
From conjugacy to the predictive. By (5), B | X1…n ~ BP(c + n, (c B0 + Σ Xi)/(c + n)). The mean of a beta process is its base measure (the paper’s appendix), and P(Xn+1 | X1…n) = E[P(Xn+1 | B) | X1…n], so Xn+1 | X1…n is a Bernoulli process with hazard (c B0 + Σi Xi)/(c + n). That measure has a discrete part, mass m_{n,j}/(c + n) on each feature ωj already held by m_{n,j} objects, and a continuous part (c/(c + n))B0 of total mass cγ/(c + n). A Bernoulli process includes each atom independently with its mass and is a Poisson process on the continuous part, which is (6).
The count. New features at step i + 1 are Poisson(cγ/(c + i)), and the Poisson draws are independent, so the number of distinct features after n objects is Poisson(γ Σ_{i=0}^{n−1} c/(c + i)), (7).
The match. At c = 1, γ = α: object i = n + 1 takes feature j with probability m_{n,j}/(n + 1) = m_j/i and Poisson(α/i) new ones, which is the buffet of LIT-1556 §4.4. The count is α Σ_{i=0}^{n−1} 1/(1 + i) = αH_n, the IBP’s E[K+] (re-computed at n = 12, α = 3: 9.3096 both ways). Since the sequence is generated by i.i.d. draws given B, B is its de Finetti measure.
What c does. Each object’s expected number of features is E[X(Ω)] = E[B(Ω)] = γ for every c, while the distinct count runs from Poisson(γ) as c → 0 (all objects share) to Poisson(nγ) as c → ∞ (none do).
What this does not say
- That features are dependent. Given B, features are independent across atoms (a Lévy process has independent increments), so the beta process carries no correlation between features. That needs another layer (LIT-1544).
- Anything about time or covariates. The objects are exchangeable.
- That the paper proves conjugacy; it cites it.