Structured mean-field inference
Mathematical Methods AIMech:StructuredMeanFieldInference DRAFT
Aliases: Structured mean-field approximation
A variational inference method that approximates a target distribution with a tractable family retaining selected dependencies, choosing its parameters to reduce the approximation-to-target Kullback–Leibler divergence.
Identity Notes & Concept Scope
This record covers the tractable-substructure mean-field construction exemplified by Saul and Jordan, including partially factorized modules and the qualifications for a tractable correlated skeleton. It is broader than the maintained factorial-HMM-specific algorithm and distinct from a fully factorized approximation that removes every intervariable dependence. It is an inference method, not a graphical-model architecture, all variational inference, a particular Monte Carlo gradient estimator or a variational autoencoder. Fully factorized mean field is a limiting restricted family when retained substructures reduce to individual variables; that inclusion does not make the two concepts exact aliases. No normalized identity match was found across all 442 pinned AIO labels/scoped synonyms and 382 preceding native IDs/names/aliases for the four researched candidates. Both full-checkout fixed-string searches included ignored and hidden files, with 45 and 45 matching lines. All 31 unique matched contents were read: prior specialized inference records, source/review context and pending leads, not exact records for the three new identities. Expanded titles, source IDs, plural forms and complete/partial-factorizability terminology were included. All 68 semantic label/synonym hits and 23 exact fully annotated AIO CLASS nodes with outgoing edges were read. Their generated YAMLs and seven native neighbors were read or byte-verified unchanged after prior complete reading, totaling 30 contextual records. The broader phrase structured variational inference is searched but not asserted as an unconditional alias of structured mean field. Boltzmann trees remains an unresolved full-text lead. Absence is bounded to the pinned vocabulary and inspected neighborhood.
Claims & Grounded Evidence
1 Claim
Mechanism
#method
For a fixed normalized target p(s)=exp(F(s))/Z and tractable q, maximize L(q)=E_q[F(s)]+H(q), so log Z-L(q)=KL(q||p). The source’s binary pairwise modular construction retains all intramodule interactions J and factorizes q across modules, replacing omitted intermodule interactions K by adjustable unary fields H. Expectations within a module are computed exactly using its tractable algorithm; expectations between modules factorize. A module’s field update is H_i^mu=h_i^mu+sum_(nu!=mu),j K_ij^(mu,nu) E_q[s_j^nu]. Holding all other factors fixed, recomputing that module with these fields gives a coordinate variational update. A chain module can use forward–backward while retaining within-chain correlations. If omitted links connect variables within the same correlated skeleton, the simple intermodule rule no longer follows: the source’s stationarity equations contain covariance and triple-moment terms. Auxiliary binary link variables provide another construction that induces a tractable core-versus-link factorization.
⚠️ Curated Limitations & Negative Results
- Tractability of retained substructures and evaluation of the required expectations must be established for the chosen family. Compact notation or a sparse-looking graph does not by itself ensure efficient inference.
- The optimized bound concerns log Z. It is an observed-data log-likelihood bound only when Z is that likelihood under the specified unnormalized posterior construction. A generic partition function is not automatically a probability of observations.
- Coordinate ascent with exact factor optimization can improve the bound; arbitrary simultaneous field updates need not. The objective can have multiple stationary points, and neither global optimality nor accurate posterior marginals is guaranteed.
- Omitted links inside one correlated substructure require the covariance/triple-moment terms of Equation (19); simply substituting independent means would change the derivation.
- The auxiliary-variable equality in Equations (20)–(21) omits a coupling-dependent constant. For one binary edge K, a=acosh(exp(2*abs(K)))/2, b=sign(K)*a and C=exp(-abs(K))/2 give exp(K*s_i*s_j)=C*sum_w exp((a*s_i+b*s_j)*w). Dropping C preserves a normalized marginal but shifts the raw log partition; evidence bounds must retain the offset.
- The context-sensitive HMM example has boundary conventions and a delayed observation term that need care; the displayed unary-field formula is not adopted as an implementation-ready recipe. Its synthetic illustration does not establish general weak-coupling accuracy or benchmark superiority.
Artificial Intelligence Ontology (AIO) Grounding
Mapped AIO Classes
| AIO Target Class | Mapping Relation | Mapping Rationale |
|---|---|---|
AIO:ProbabilisticGraphicalModel
Probabilistic Graphical Model
|
related_match | The method approximates distributions whose dependence structure is represented by a graph. The pinned target denotes a graphical-model task, while this local concept denotes a variational inference method; the relation is method-to-modeling context, not an is-a assertion. |
AIO:BoltzmannMachineNetwork
Boltzmann Machine Network
|
related_match | The primary derivation approximates the binary pairwise Boltzmann distribution and its hidden-variable statistics. The AIO target is the probabilistic network family, not the approximation procedure. This direct method-to-model relation does not restrict the method to one trained network. |
Evidence Sources & Primary Literature
1 Source| Source ID | Publication / Document | Type / Access | Reference & Checksum |
|---|---|---|---|
method-source
|
Lawrence K. Saul, Michael I. Jordan
|
primary paper
full_text
|
ID: https://proceedings.neurips.cc/paper/1995/hash/285f89b802bcb2651801455c86d78f2a-Abstract.html
sha256:d0a9f10a9d70…
|
Curation Provenance
Curator: Codex
Method: ai_assisted
Model: GPT-6
Created: 2026-10-11T20:15:57Z
Updated: 2026-10-11T20:15:57Z
Source File:
data/curated/StructuredMeanFieldInference.yaml