Source
Griffiths and Ghahramani (2011), JMLR 12: 1185–1224, LIT-1556, read in full in NOTE-1172. The finite model is their (9), the per-feature expectation their (11), and K+ ~ Poisson(αH_N) their §4.6, stated for the limit. The finite-K count and the inequality below are the record’s.
What was actually shown
The model: for k = 1 … K, πk ~ Beta(α/K, 1) independently, and zik | πk ~ Bernoulli(πk) for i = 1 … N.
Per object. E[zik] = E[πk] = (α/K)/(1 + α/K), so an object’s expected number of features is K(α/K)/(1 + α/K) = Kα/(K + α). The paper gives the total, Nα/(1 + α/K), and its bound Nα (11).
Occupied features. Column k is occupied unless all N entries are 0:
P(mk > 0) = 1 − E[(1 − πk)^N] = 1 − (α/K)∫₀¹ π^{α/K − 1}(1 − π)^N dπ = 1 − (α/K)B(α/K, N + 1).
The columns are identically distributed, so E[K+] = K(1 − (α/K)B(α/K, N + 1)).
Below αH_N. Write ε = α/K. Then εB(ε, N + 1) = Γ(1 + ε)Γ(N + 1)/Γ(N + 1 + ε) = ∏_{j=1}^{N} j/(j + ε) = ∏_j (1 − xj), with xj = ε/(j + ε). By the Weierstrass product inequality ∏(1 − xj) ≥ 1 − Σ xj, so
E[K+] = K(1 − ∏(1 − xj)) ≤ K Σ_j ε/(j + ε) < K Σ_j ε/j = αH_N,
strictly for every finite K. As K → ∞, ∏ j/(j + ε) = 1 − εH_N + O(ε²), so E[K+] → αH_N, the paper’s limit.
Numbers (re-computed 2026-10-26, exact formula). N = 12, α = 3: αH₁₂ = 9.310. E[K+] is 6.830 at K = 16, 8.381 at K = 50 and 9.259 at K = 1000. Per object, Kα/(K + α) is 2.53 at K = 16 against α = 3. Under the IBP itself P(K+ > 16) = 0.0149. For comparison, a stick-breaking truncation at 16 (Teh et al. 2007, the construction LIT-1548 uses), with vj ~ Beta(α, 1) and bk = ∏{j≤k} vj, gives 8.97 occupied features (Monte Carlo, 200,000 draws) and Σ{k≤16}(α/(1 + α))^k = 2.97 features per object.
What this does not say
- That the finite model is a poor prior. It is a valid exchangeable prior in its own right; it is not the IBP with α at that K, and the IBP’s quantities should not be quoted for it.
- That truncation must be large. A stick-breaking truncation at the same K is close to the IBP, because it keeps the largest prevalences and drops only the tail; the finite beta–Bernoulli model spreads α over K exchangeable columns instead.
- Anything about the posterior. The shortfall is in the prior; with data that need more features than the prior expects, the posterior can still occupy them, at a prior cost.
Note of 2026-10-27: the stick-breaking comparison, at its source
The Numbers above compare the finite model with a stick-breaking truncation at 16, cited as Teh et al. 2007 and computed by Monte Carlo. That paper was registered and read on 2026-10-27, at the owner’s request (“file the next readings”), as LIT-1627 (NOTE-1210). From the construction the expected count has a closed form, and at K = 16, N = 12 and α = 3 it is 8.974, with 2.97 features per object (THEORY-1041), which confirms both figures.
“What this does not say” explains the truncation’s closeness by its keeping “the largest prevalences and drop[ping] only the tail”. The tail is not empty: in 24% of draws at least one feature beyond the sixteenth stick is occupied, and their expected number, 0.34, is the whole gap 9.31 − 8.97. The comparison stands. The IBP’s original paper, LIT-1623, read the same day, states αH_N for the limit only, as the review does (THEORY-1043). The title and status are unchanged.