Source
Williamson, Orbanz and Ghahramani (2010), AISTATS, PMLR 9: 924–931, LIT-1548, read in full in NOTE-1175; equations (1), (3)–(5) and Appendix A. The paper states “For each t, the marginal distribution of Z(t) is an IBP” (§3) and “has IBP marginals at any covariate value” (footnote 1). The matrix-level analysis below is the record’s.
What was actually shown
Sticks: vj ~ Beta(α, 1), bk = ∏_{j≤k} vj, the IBP’s stick-breaking construction, not depending on t. Model (4): gk ~ GP(0, Σk), hnk ~ GP(gk, ρ²I), znk(t) = I{hnk(t) < F⁻¹(bk | 0, Σk(t,t) + ρ²)}.
Per item, prevalence is fixed. Integrating out gk, hnk(t) is N(0, Σk(t,t) + ρ²), so P(znk(t) = 1 | b) = bk at every t. Each row of Z(t) is therefore distributed as an IBP row, holding Poisson(α) features in the limit, at every t. Persistence is a separate matter: for a stationary process with correlation r between hnk(t) and hnk(t + 1), P(znk(t + 1) = 1 | znk(t) = 1) = Φ₂(c, c; r)/bk with c = Φ⁻¹(bk).
The matrix, collapsed case. With Σk = 0 the hnk are independent across n, so given b the rows of Z(t) are independent with P(znk = 1) = bk. That is the stick-breaking IBP exactly.
The matrix, shared case. With Σk(t,t) > 0, given gk(t) the items are independent Bernoulli(γk(t)), γk(t) = Φ((c̃k − gk(t))/ρ), and γk(t) is random with mean bk. So for two items E[z1k z2k | b] = E[γk(t)²] > bk²: they are positively correlated. Since (1 − x)^N is convex, E[(1 − γk)^N | b] ≥ (1 − bk)^N by Jensen, so each column is less likely to be occupied than under the IBP. Simulated at N = 12, α = 3, ρ² = 1, 60 sticks, 40,000 draws: E[K+] = 9.34 with Σk(t,t) = 0 (IBP: αH₁₂ = 9.31), 7.06 with Σk(t,t) = 1 and 5.35 with Σk(t,t) = 4. Mean features per row 3.00 in all three.
Persistence and prevalence (exact bivariate normal, 2026-10-26). At r = 0.5, 0.9 and 0.99 a held feature is kept with probability 0.18, 0.59 and 0.87 when bk = 0.025, and 0.44, 0.75 and 0.92 when bk = 0.2.
What this does not say
- That the paper’s prediction results depend on the matrix marginal. Its inference is truncated Gibbs on the model as defined, whatever its marginal law.
- That the dIBP makes features depend on one another or on the other items’ features. Given the sticks and the GPs, columns are independent.
- That prevalence can change over t. It can only through the shared gk, and then only at the cost above.
- That one covariance holds rare and common features alike. For a given correlation, rare features are less persistent; per-feature length scales, which the paper uses, are what make persistence a per-feature parameter.