Source
Van Gael, Teh and Ghahramani (2008), The infinite factorial hidden Markov model, LIT-1577, read in full in NOTE-1199: the finite model (3)–(4), from which this is derived. Williamson, Orbanz and Ghahramani (2010), LIT-1548, footnote 1 (NOTE-1175), states the conclusion without derivation.
What was actually shown
The derivation. A two-state chain with P(0 → 1) = a and P(1 → 1) = b, started off, is on at step t with probability π₁(1 − λ^t), where π₁ = a/(1 − b + a) and λ = b − a. With M features and a_m ~ Beta(α/M, 1), M·E[g(a)] → α ∫₀¹ g(a)/a da for any g with g(0) = 0, the IBP’s limit, since Beta(α/M, 1) has density (α/M)a^{α/M − 1}. So, with b independent of a,
E[number on at t] → α ∫₀¹ E_b[(1 − (b − a)^t)/(1 − b + a)] da.
At t = 1 the integrand is 1, giving α, the IBP’s Poisson(α) mean for one row. As t → ∞ the limit is α ∫₀¹ E_b[1/(1 − b + a)] da = α E_b[ln((2 − b)/(1 − b))]. That is finite for every γ, δ > 0, since E[−ln(1 − b)] = ψ(γ + δ) − ψ(δ).
The values. At α = 3:
| (γ, δ) | t = 1 | t = 2 | t = 10 | t = 50 | t = 245 | limit |
|---|---|---|---|---|---|---|
| (10, 1), the paper’s prior means | 3.00 | 4.23 | 7.03 | 8.50 | 8.92 | 9.04 |
| (10, 3) | 3.00 | 3.81 | 5.20 | 5.41 | 5.42 | 5.42 |
| (1, 1) | 3.00 | 3.00 | 3.87 | 4.10 | 4.15 | 4.16 |
Simulation of the finite model (M = 20,000, 200 replicates) gave, for (10, 1), 2.90, 4.32, 7.26, 8.77 and 9.20 at the same steps, and agreed to within about 3% for the other settings (record simulation, 2026-10-27).
What the paper does say. The number of chains ever on, M₊, is Poisson(αH_T) (§2.3). That is consistent: it counts columns ever used, not columns on at a step.
What this does not say
- Not the full distribution of a row after the first step. Only the expected count is derived; the count at a step is Poisson in the limit (a Poisson process thinned independently), but nothing more is claimed.
- Not a defect relative to the paper’s purpose. The paper wants features that persist (“people speak for larger stretches of time”), and claims no IBP marginal per row.
- Not about several objects. The model has one sequence of rows.
Connections
- CASE-165, item 2. The case finds that T23’s sticky term holds the buffet’s prior properties “at t = 0 only”. The Markov IBP has the same property for the count of active features: α at the first step, three times α later at the paper’s own prior means. Neither the sticky term nor its closest established relative keeps the prior’s marginal over time. Of the IBP’s dependent variants in the record, only the dependent IBP does (THEORY-1006).
- THEORY-999. That account concerns a finite truncation’s count of occupied features against the limit’s αH_N. This one concerns the count at a time step against the IBP’s α. Both are cases of an IBP figure quoted where it does not hold.