CASE-165: A dynamic beta–Bernoulli latent-feature model of an organization's members: a finite Indian buffet prior over who holds which interpretive features, a sticky logistic transition for holding them, activations, and team, communication and authorization layers above; at the proposed size of sixteen features it falls well short of the Indian buffet process it quotes, its persistence term does not hold rare features, and the difference between having a capacity and using it is not identified from the outputs

The construction (T23)

For employees e = 1 … N, latent features k = 1 … K and days t:

• Prior (T23 §3). π_k ~ Beta(α/K, 1), B_{ek,0} | π_k ~ Bernoulli(π_k), W_k ~ p(W). “As K grows, the induced distribution over the initial binary feature allocations approaches the familiar one-parameter IBP construction”, with “E[K_N] = αH_N”.
• Holding a feature (§4), “a sticky logistic transition”: logit p_{ek,t+1} = logit(π_k) + ρB_{ek,t} + β_social Σ_f C_{ef,t}B_{fk,t}
  • β_team Σ_g M_{eg,t} B̄_{gk,t} + β_learn r_{ek,t}, and B_{ek,t+1} ~ Bernoulli(p_{ek,t+1}). T23 names it accurately: “an IBP-inspired initial allocation with a separately specified dynamic transition model. It is not automatically the exact dependent IBP of Williamson, Orbanz, and Ghahramani”.
• Using a feature (§5). An activation a_{ek,t} ∈ [0, 1], Beta-distributed, and the interpretation u_{e,t} = Σ_k B_{ek,t} a_{ek,t} W_k o_{e,t} + W_e^private o_{e,t}, with optional pairwise terms. “Feature availability should not be confused with feature use.”
• Access, world and reports (§5–6). A signal reaches an employee with a probability set by team, role, communication and attention; the world is X_{t+1} = f(X_t, A_t) + ε_t; reports are O_{j,t} ~ N(H_j X_t + b_j(ρ_{j,t}), σ_j²), with a reporting bias the organization can move.
• Organization (§7–8). Poisson communication attempts with typed messages, team records L_{g,t}, proposals by softmax over local appraisals, and enactment by P(A_t | P_t, Γ_t, L_t, C_t) through a permission structure Γ_t. The established form of a team layer at t = 0 is the hierarchical beta process (THEORY-1015): each team’s prevalence of a shared feature is drawn around a shared one. It puts each member in one group, fixed, so it cannot give the overlapping and changing membership M_{eg,t} or the β_team dynamics. “A frequently active employee may have little authority. An employee who rarely communicates may possess a critical veto.”
• No perspective variable (§10). “It does not contain a primitive random variable whose value is the organization’s current perspective.” A perspective is “a derived hypothesis about a persistent regime” and a tuple of features, relations, governance and history, inferred afterwards. “The IBP prior supplies multiplicity; the relational dynamics will have to supply organization” (§3). That division is CLAIM-454.
• Proposed size (§12, “Northstar generative study v0”). N = 12, 4 formal teams, K = 16, α = 3, 5 world dimensions, 64 days: “proposed experimental settings, not measured results”.
• How to use it (§16). “Option A — Mechanistic simulator + IBP inference” first: fit the feature model to a simulator that does not assume it, so that “feature discovery can be tested against a world that did not itself assume IBP-distributed employees”.

What checks out

• The IBP’s sequential construction (T22) and E[K_N] = αH_N. For N = 12 and α = 3, αH₁₂ = 9.310, and simulating the buffet gives 9.29 to 9.33. The beta process gives the same count a second way, as the IBP’s de Finetti measure (THEORY-985).
• The finite beta–Bernoulli limit. E[occupied features] is K(1 − (α/K)B(α/K, N + 1)), which is 8.38 at K = 50 and 9.26 at K = 1000 and tends to αH_N, from below for every finite K (THEORY-999).
• The joint factorization (§9) is a valid chain-rule factorization under its stated Markov assumptions. It does let decisions depend on the hidden world directly, p(D_t | O_t, S_t, Θ), so “access comes before interpretation” is intended and not enforced.
• The two non-identifiabilities §15 names: column permutation, and the scale trade of W_k against a_{ek} (exact while the rescaled activation stays in [0, 1]). §15 also says that “Closely related latent features may also split or merge without significantly changing the likelihood”.

What does not

1. The truncation is not generous at the size chosen. T22 recommends “a finite, generously truncated beta–Bernoulli approximation”; T23 fixes K = 16 and quotes αH_N. At K = 16 the expected number of occupied features is 6.83 (simulated 6.82), 27% short of 9.31, and the expected number per employee is Kα/(K + α) = 2.53, not α = 3. The buffet itself seldom needs more than 16 (P(K_N > 16) ≈ 0.015), so a stick-breaking truncation at 16 would be close to it (8.97, simulated). The finite model at 16 is not. The review T22 and T23 cite states αH_N for the limit only (§4.6); for finite K it gives no count of occupied features, so the quotation at K = 16 has no support there (THEORY-999, NOTE-1172).
2. The sticky transition does not hold rare features. Without social, team or learning input, a held feature is kept with probability σ(logit π_k + ρ). Under the prior at K = 16, E[ln π_k] = −K/α = −5.33 and the median π_k is 0.025, where the probability of keeping a held feature is 0.16 at ρ = 2, 0.58 at ρ = 4 and 0.91 at ρ = 6. “A feature persists because an employee retains it” holds only where ρ is near −logit π_k, and one ρ cannot do that for features of different prevalence. The stationary prevalence also departs from π_k (0.36 against 0.2 at ρ = 2, 0.75 at ρ = 4), so the buffet’s prior properties hold at t = 0 only. The established relatives fix what this term does not. The dependent IBP T23 names thresholds a Gaussian process at the quantile of a stick that does not change with time, so each feature’s prevalence stays fixed and its persistence is set separately (THEORY-1006); but for one correlation a rarer feature is still kept less often (0.59 at a prevalence of 0.025 against 0.75 at 0.2, for correlation 0.9), and only per-feature length scales make it a per-feature parameter. The Markov IBP (Van Gael et al. 2009, described in the review’s §7.4, not held) gives each feature its own probabilities of turning on and staying on. And with team-specific prevalences, as in a hierarchical beta process (THEORY-1015), one ρ fails across teams as well as across features.
3. Having and using are not identified from the outputs. §2 calls the separation of B and a “important” (“An engineer may possess the capacity to interpret customer harm but not activate that interpretive mode”). In u = Σ B a W o + W^private o:
   • B = 1 with a near 0 and B = 0 give nearly the same output, so availability is told from activation only by the prior and the transition, not by the data.
   • A feature one employee holds at constant activation c is absorbed into that employee’s private map, W^private + cW_k, exactly.
   • Features whose B and a columns coincide across employees are identified only through W_k + W_l, exactly. These are instances of CLAIM-148’s gauge, which says a factorization is identified only up to an invertible change of latent coordinates. A correlated prior over features does not remove the third: it separates co-occurring features only where they sometimes occur apart (THEORY-1012, NOTE-1184). The author–topic model has the absorption of the second in published form. Its per-document “fictitious author” plays W^private, a topic used by one document can sit there or in a real author’s distribution as the priors choose, and adding it improves prediction throughout (THEORY-1003).
4. Splits and zero contributions are free. In the per-component form T24 ports to transformers, r_c(x) ≈ Σ_k B_ck a_k(x) v_ck, rescaling a feature is exact, splitting a_k into two columns with copied B and v raises K by one at no cost, and B_ck = 1 with v_ck = 0 equals B_ck = 0. A generic invertible mix of features destroys all 15 zeros of B∘V in a 6 × 4 check, so sparsity is what identifies features, up to permutation, scale and splits. A “birth” in B can therefore be an artefact of the prior or of a split, and needs a test beyond reconstruction, such as a causal effect or CLAIM-263’s quotient by the full symmetry group. The review’s own worked example shows both freedoms: in its photograph demonstration three of the four main features code an object’s absence, and three extra ones code “slight differences in the locations” (NOTE-1172), so a 1 in a column need not mean “has” and a feature can be a split of another.

All of these were re-computed when this case was filed (exact formulas, and Monte Carlo over 20,000 draws or more).

The port to transformer checkpoints (T24)

T24 carried the model to components c of a transformer across checkpoints s, with a temporal prior p(B_s | B_{s−1}, θ_s, 𝒟_s), “where 𝒟_s summarizes training exposure”. Two further gaps:

• The prior conditions on what it should predict. B_s is inferred from activations fixed by θ_s, so conditioning on θ_s leaves the prior nothing to forecast. The prospective test T24 itself asks for, fitting early checkpoints and predicting later ones, needs B_{s+1} predicted from B_{≤s}, exposure and intervention without θ_{s+1}. Since the true dynamics are known ((θ, m) updated by the optimizer), a dynamic model of B is a coarse-graining, and whether it is adequate is a closure question (TERM-068, QUESTION-252).
• Components are not exchangeable. The buffet treats its customers as exchangeable. Heads and MLP channels are distinguished by layer and by what they read and write, so a layer-indexed or hierarchical prior would be needed.

In a transformer the third gap above, having against using, is measured by intervention rather than left to the prior. Cross-checkpoint crosscoders separate a feature’s presence at a checkpoint (its decoder norm) from its causal use on a task (its indirect effect), and a feature can be present and idle (NOTE-1186, THEORY-1004). The refined-LLC study compares a head’s weight sensitivity with output ablation, which mostly agree and sometimes do not (NOTE-1188). A crosscoder’s jointly trained dictionary is the deterministic analogue of this model’s allocation without a temporal prior, and the split caution of item 4 applies to it too. Where a task’s symmetries fix the components’ features, as in Musat, Pimentel, Zucchet and Hofmann’s idealized attention-only model (THEORY-981), both identifiability worries are removed by construction: the coordinates are given, not inferred.

What it can show

What a latent-feature model of members must get right before its features can be read as anything: a truncation that matches the prior quoted, a persistence term scaled to prevalence, and a test that separates having from using and a birth from a split. It is the form in which QUESTION-276’s candidate method would be judged, and its §10 division is CLAIM-454’s.

What it cannot show

That any firm’s or network’s members hold such features, or that the model fitted to anything recovers a perspective. Nothing has been fitted. “An IBP prior by itself cannot produce organizational capture, self-maintaining coalitions, or effective institutional learning” (T23 §13).

Note of 2026-10-27: item 2 and the Markov IBP, now held

Item 2 names “The Markov IBP (Van Gael et al. 2009, described in the review’s §7.4, not held)”. It was registered and read on 2026-10-27, at the owner’s request (“get started on the seven works and markov IBP”), as LIT-1577: Van Gael, Teh and Ghahramani, The Infinite Factorial Hidden Markov Model (NOTE-1199), the paper that introduces the Markov Indian buffet process. It appeared at NIPS 21, the conference of December 2008; the review (LIT-1556) cites it as 2009, the year of the printed volume. It is the same work, and the record dates it 2008-12-01.

• What it gives each feature (THEORY-1034). Each feature’s probability of staying on, b ~ Beta(γ, δ), is drawn independently of its probability of switching on, a ~ Beta(α/M, 1), so persistence does not depend on rarity. T23’s sticky term, with its other inputs at zero, is the same two-state chain per feature, with on-rate π_k and the stay-rate tied to it as σ(logit π_k + ρ). At the paper’s prior means every feature, however rare, is kept with E[b] = 10/11 = 0.91; T23 needs ρ = 5.97 to reach that for the median feature (π = 0.0248), and then mis-scales common ones. So the Markov IBP fixes the coupling of persistence to rarity that item 2 finds.
• What it does not fix (THEORY-1019). Its expected number of features on is α at the first step but rises toward α·E[ln((2 − b)/(1 − b))], 9.04 at the paper’s prior means (γ = 10, δ = 1, α = 3), three times α; after the first step the rows are not IBP-distributed. Prevalence is not held either: the stationary on-fraction is a/(1 − b + a), 0.22 at a = 0.025 with b = 10/11. So the “t = 0 only” problem item 2 names remains under the Markov IBP; of the relatives the record holds, only the dependent IBP keeps prevalence fixed over time (THEORY-1006).
• The members’ axis. “Gives each feature its own probabilities” holds for one sequence of time steps. The model has no axis of members, and sharing (a, b) across N members is not derived in the paper.
• Item 2’s numbers re-check. The keep probabilities 0.16, 0.58 and 0.91 at ρ = 2, 4 and 6 for the median π, and the stationary prevalences 0.36 and 0.75, which are the same formula a/(1 − b + a) with b = 0.649 and 0.932 (NOTE-1199).

The paper’s printed predictive rules for its sequential buffet are both misprinted; the record’s reading uses the rules derived from its eq. (6) (NOTE-1199). The case is unchanged.

Note of 2026-10-27 (later): the IBP’s originals and the factorial HMM

Five works the infinite factorial HMM names were registered and read on 2026-10-27, at the owner’s request (“file the next readings”). Four bear on this case’s items, and one on the model’s form.

• Item 1: the 8.97 is exact, and its reason is looser than stated. Teh, Görür and Ghahramani’s stick-breaking construction, LIT-1627 (NOTE-1210), gives the expected number of occupied features under a truncation at K sticks in closed form (THEORY-1041). At K = 16, N = 12 and α = 3 it is 8.974, and a simulation of 200,000 draws gives 8.969 ± 0.006; P(K_N > 16) = 0.0149 is confirmed; per employee the count is 2.97, not 3. Item 1 says that because the buffet “seldom needs more than 16”, a stick-breaking truncation at 16 “would be close”. What a truncation loses is the features beyond the sixteenth stick, which is not the same event: in 24% of draws at least one of them is occupied, and their expected number, 0.34, is exactly the gap 9.31 − 8.97. The conclusion “close” (3.6% short, against the finite model’s 27%) stands. At the same K = 16, the finite model (6.83, THEORY-999) and the truncated construction (8.97) are different priors, and T23’s is the finite one.
• Item 3: “not by the data” holds with the activation fixed. Knowles and Ghahramani’s infinite sparse factor analysis, LIT-1626 (NOTE-1209), has the same mask-times-value form as B and a. With the value integrated out under its prior, the evidence of a data point for “active” is σ̂·exp(μ̂²/(2σ̂²)), and where the residual has no component along the loading it is σ̂ < 1: the data then favour “not held” over “held and unused”, by a factor the prior on the value sets (THEORY-1046). So availability is told from activation by the data as well, weakly and only through the prior on a. The paper’s own recovered mask shows gaps where the value is zero. Item 4’s “B_ck = 1 with v_ck = 0 equals B_ck = 0” is the same identity and takes the same qualification under a prior on v. The confounds item 3 lists between features and the private map are untouched.
• Item 4: the worked example is the original’s. The photograph demonstration item 4 takes from the review, three of four main features coding an object’s absence and three coding “slight differences in the locations”, is first reported in the same words in Griffiths and Ghahramani’s NIPS paper, LIT-1623 (NOTE-1208), §3 and the caption of its Figure 3. The point stands on the original.
• The model per member is a factorial HMM. With each B_ek a two-state chain, u_e = Σ_k B_ek a_ek W_k o + W_e^private o is, for one employee, a factorial hidden Markov model with K binary chains: Ghahramani and Jordan, LIT-1628 (NOTE-1203), the parametric original that LIT-1577 makes nonparametric. Two things follow. Given the outputs the chains are coupled, which is how item 3’s confounds arise. And exact inference costs O(TMK^{M+1}) for M chains of K states: at sixteen binary features, 16·2¹⁷ ≈ 2.1 million terms per employee per day. A fit needs the paper’s structured variational approximation or a whole-chain sampler of the beam type (Van Gael, Saatci, Teh and Ghahramani, LIT-1622, NOTE-1204), whose exactness is THEORY-1037. The case’s sticky chains with weakly informative outputs are the regime that paper reports as hardest for single-site Gibbs sampling; its claim of faster mixing is THEORY-1044, Proposed. The reading also corrects a footnote of Ghahramani and Jordan’s: orthogonal features decouple the chains given an observation only under isotropic noise, not under any diagonal covariance.

The case is unchanged: its title, standing and status stand.

Note of 2026-10-30: what holds it back

The title’s third clause, that having a capacity and using it are “not identified from the outputs”, is qualified by this case’s note of 2026-10-27 (later): under a prior on the activation the data favour “not held” over “held and unused”, weakly. The clause needs narrowing to the confounds that are exact (absorption into the private map, co-occurring features identified only in sum) before the case can be promoted.