Source
Griffiths and Ghahramani (2005), Infinite latent feature models and the Indian buffet process, LIT-1623, read in full in NOTE-1208: §2.1 (the finite model, (1)–(2)), §2.2 (lof-classes), §2.3 (the limit, (3)–(4)) and §2.4 (the buffet, (5)). The derivation is given at length in the authors’ review, LIT-1556 (§§3–4, Appendix A; NOTE-1172).
What was actually shown
The finite model. With z_ik | π_k ~ Bernoulli(π_k) and π_k ~ Beta(α/K, 1), P(Z) = ∏_k (α/K)Γ(m_k + α/K)Γ(N − m_k + 1)/Γ(N + 1 + α/K), and the probability of a lof-class multiplies it by K!/∏_h K_h!.
The limit. As K → ∞, P([Z]) → α^{K+}/∏{h=1}^{2^N − 1} K_h! · exp{−αH_N} · ∏{k=1}^{K+} (N − m_k)!(m_k − 1)!/N!. It depends only on the K_h and m_k, so it is infinitely exchangeable in the objects.
The buffet. The first object takes Poisson(α) features; the i-th takes each earlier feature with probability m_k/i, then Poisson(α/i) new ones. Its matrices map to lof-classes with the probabilities of the limit. The new features over N objects number Poisson(Σ_i α/i) = Poisson(αH_N), and each object holds Poisson(α).
The check. At N = 3 and α = 1.5, for the six commonest lof-classes, the finite model at K = 100,000, the limit and 400,000 buffet runs give: the empty matrix 0.0639, 0.0639 and 0.0634; one feature held by the first object only 0.0320, 0.0320 and 0.0325; one held by all three 0.0320, 0.0320 and 0.0322; one held by the second only 0.0320, 0.0320 and 0.0321; one held by the third only 0.0320, 0.0320 and 0.0318; one held by all and one by the first only 0.0160, 0.0160 and 0.0162. The simulated mean of K+ is 2.751; αH_3 = 2.750 (record script, 2026-10-27).
What this does not say
- Nothing about finite K. The count αH_N is the limit’s. The finite model expects K(1 − (α/K)B(α/K, N + 1)) occupied features, strictly fewer (THEORY-999).
- Nothing about a posterior. The result is about the prior; the paper’s sampler, and the review’s footnote that its posterior version is heuristic, are separate matters (NOTE-1172).
- Not that a column means “has”. The likelihood decides what a feature codes; in the paper’s own demonstration three of four main features code an object’s absence (NOTE-1208).
Connections
- CASE-165, item 1. The case’s αH₁₂ = 9.31 for α = 3 is this limit’s mean, and its correction for K = 16 is THEORY-999’s.
- THEORY-985. The beta process gives the same count a second way, as the IBP’s de Finetti measure.
- THEORY-1041. Ordering the limit’s feature probabilities gives the stick-breaking construction, whose truncations approximate this limit from below.
- THEORY-1019. The Markov IBP repeats this argument with rows as time steps, and keeps this marginal only at the first step.