Source
Airoldi, Blei, Fienberg and Xing (2008), JMLR 9:1981–2014, LIT-1553, read in full in NOTE-1179 from the JMLR copy. The model is §2 and §2.1, the estimators §3.2 and Appendix B. The paper’s own statement, in §5: B and the memberships are “apparently in competition for explaining the data, possibly raising an identifiability issue. This is not the case, however, as the blockmodel B captures global/asymmetric relations, while the mixed membership vectors Πs capture local/symmetric relations. This difference practically eliminates the issue, unless there is no signal in the data to begin with.” The derivations below are the record’s.
What was actually shown
The edge probability. For each ordered pair, z_{p→q} ~ Mult(π_p) and z_{p←q} ~ Mult(π_q) independently, and Y(p, q) ~ Bernoulli(σ_pq) with σ_pq = (1 − ρ) z_{p→q}ᵀ B z_{p←q}. Summing over the K² values of the two indicators gives P(Y(p, q) = 1 | π_p, π_q) = (1 − ρ) Σ_{g,h} π_p(g) B(g, h) π_q(h) = (1 − ρ) π_pᵀBπ_q. The indicators are drawn afresh for each pair, so given the memberships the edges are independent (Appendix A), and the likelihood of the graph given Π is Π_{p,q} of Bernoulli terms in (1 − ρ)π_pᵀBπ_q.
The gauge. For invertible R, (Rπ_p)ᵀ(R⁻ᵀBR⁻¹)(Rπ_q) = π_pᵀBπ_q. So (RΠ, R⁻ᵀBR⁻¹) gives the same likelihood given memberships whenever it is a valid parameter: each Rπ_p on the simplex and R⁻ᵀBR⁻¹ in [0, 1]. Permutations always qualify, and the paper uses that freedom itself when it “estimate[s] a permutation” of blocks to functions (§4.3.2). A continuum qualifies too. Take R = (1 − ε)I + (ε/K)J. It is nonnegative with columns summing to one, so it maps the simplex into itself. And R⁻ᵀBR⁻¹ = B + O(ε), so it stays in [0, 1] for small ε when every rate of B is strictly between 0 and 1. The change moves every membership toward the simplex’s centre and sharpens B to compensate.
The paper’s reason does not block this. R⁻ᵀBR⁻¹ is still one global, possibly asymmetric matrix, and Rπ_p is still one vector per node, used for both directions of its edges. The split between “global/asymmetric” and “local/symmetric” is kept by the transformation.
What can still separate them. If π_p ~ Dirichlet(α), then Rπ_p is not Dirichlet-distributed for a non-permutation R. So the marginal likelihood p(Y | α, B, ρ), with the memberships integrated out, is not shown to be invariant, and (α, B) may be identified through the Dirichlet family. If so, the identification is the prior’s shape doing the work, not the graph.
ρ and B’s scale. σ_pq depends on ρ and B only through (1 − ρ)B, before any marginalization. So for c > 1, (ρ′, cB) with 1 − ρ′ = (1 − ρ)/c gives the same distribution of the graph, in the likelihood given memberships and in the marginal likelihood alike, as long as cB ≤ 1. ρ is therefore fixed only by the bound: any ρ with 1 − ρ ≥ max_{g,h} (1 − ρ)B(g, h) fits equally well.
The printed estimator. ρ̂ = Σ_{p,q} (1 − Y(p, q))(Σ_{g,h} φ_{p→q,g} φ_{p←q,h}) / Σ_{p,q} Σ_{g,h} φ_{p→q,g} φ_{p←q,h}. Each φ vector is normalized to sum to one, so the inner sum is 1 for every pair, and ρ̂ is the fraction of ordered pairs with no edge, whatever the fitted φ. Footnote 9 says this equals the density-based ρ̃ = 1 − d̂ “in the case of single membership”. As printed, they are always equal.
Checks. With N = 6, K = 3, random memberships, B uniform on [0.1, 0.9] and ε = 0.05: the transformed memberships summed to one and stayed positive, R⁻ᵀBR⁻¹ stayed between 0.177 and 0.910, and every edge probability changed by at most 2.2 × 10⁻¹⁶. The marginal of the per-interaction indicators matched π_pᵀBπ_q. The ρ–scale trade was exact to 6 × 10⁻¹⁷. For random normalized φ, the printed ρ̂ equalled the fraction of zeros.
What could have failed: the per-interaction indicators could have tied the memberships to B in a way the bilinear form does not show. They do not, because each pair’s indicators are summed out independently.
What this does not say
- Not that MMSB is unidentified in its marginal likelihood. The Dirichlet assumption may identify (α, B) up to permutation. That is open, and would be shown by a theorem on the Dirichlet family or a counterexample.
- Not that the paper’s fits are wrong. Its predictions of edges depend on π_pᵀBπ_q, which the gauge leaves alone. The readings of blocks and of how mixed a node is are what it touches. The school network’s “few students display mixed membership” is a statement about one point of the orbit.
- Not about the de-noising use with the indicators kept. With the indicators Z fixed, the complete-data likelihood is not invariant under R, since Z is an assignment and not a vector to transform.
Connections
- CLAIM-148. This is its gauge in bilinear form, X = ΠᵀBΠ, with the simplex and [0, 1] in place of nonnegativity. Like the NMF case, the freedom is data-dependent and larger than the permutations.
- THEORY-972. The author–topic model’s gauge between topics and author profiles is the same kind, with the authorship matrix known.
- CLAIM-135. The paper’s readings of blocks as factions, grades and protein functions reach the first achievement and, through the Dirichlet family, perhaps the second. §5 claims the second without an argument.
- QUESTION-276. A mixed-membership blockmodel of the communication and authority graph is the baseline its condition 4 needs. Readings of how far each member takes part in each block of that baseline inherit this gauge, so any comparison should be on predictions of edges and outcomes, not on memberships.