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Gaussian message conflict diagnostic

Algorithms AIMech:GaussianMessageConflictDiagnostic DRAFT

A Bayesian GLMM screening procedure that separates Gaussian prior and likelihood natural-parameter messages at local variational fixed points and computes conflict scores from an approximately independent Gaussian difference. It approximates predictive-prior versus unit-likelihood disagreement using full-data fitted factors.

Identity Notes & Concept Scope No normalized identity match across 442 pinned AIO and 556 maintained IDs, names, aliases and scoped synonyms for the proposed Gaussian-message and general cross-validated conflict diagnostics. Fixed-string searches of the recovery, older discovery and original Documents checkouts included ignored and hidden files and returned 59, 0 and 0 lines. All 59 were assessed: prior source links, the stochastic fitting record and its generated/review mirrors, and explicit pending diagnostic leads. Assessed 50 semantic AIO rows and 177 native names/definitions, reusing 31 and 130 verified unchanged rows and reading all 19 and 47 additional rows. Ten complete AIO nodes, annotations, incident edges and generated records were read or verified unchanged; four full native context records were read or verified against prior readings. The source phrase mixed messages diagnostics denotes this Gaussian approximation to prior-likelihood conflict scoring, not the separate GLMM fitting algorithm. It is distinct from generic NCVMP, the full-data and stochastic GLMM fitting procedures, distance-based local outlier factors and the simulation-based cross-validated comparator. This is a descriptive AIMech algorithm identity bounded to the inspected source and corpus.

Claims & Grounded Evidence

7 Claims

Scope #fitted-glmm-context

The inspected diagnostic applies to the partially noncentered Poisson-log or Bernoulli-logit GLMM with a_i=u_i+B_i*beta and predictor eta_i=V_i*beta+Z_i*a_i, where a_i|beta,D~N(B_i*beta,D). Its input is a fitted block-factorized approximation q(beta)q(D)product_i q(a_i), with q(beta)=N(m,Q), q(D)=IW(nuq,Sq) and q(a_i)=N(m_i,Q_i). Require proper factors, finite Gaussian moments and a local Gaussian NCVMP fixed point for every assessed unit at the final fixed global factors. Stochastic global fitting alone can leave local factors stale; the source calls for a final local refresh. The diagnostic compares prior and likelihood messages about unit random effects; it does not itself fit the model or prove that a flagged unit is erroneous.

supports Source: A stochastic variational framework for fitting and diagnosing generalized linear mixed models · Locator: Sections 5–5.2, PDF pp. 17–21; final local refresh, PDF p. 21

Mechanism #prior-message

Write A=E_q[D^-1]=nuq*Sq^-1. The Gaussian prior message has precision A, mean m_prior=B_i*m and covariance C_prior=A^-1=Sq/nuq. These natural parameters follow from the a_i-dependent part of E_q(beta,D)[log p(a_i|beta,D)]. After normalization it is a Gaussian function of a_i. It is not the arithmetic predictive mixture integral p(a_i|beta,D)q(beta)q(D) d beta dD: when nuq>r+1 that mixture has covariance Sq/(nuq-r-1)+B_i*Q*B_i^T, generally different. A pointwise Jensen bound on an unnormalized expected-log kernel does not establish a density lower bound after separate normalization. The mixture-moment comparison is curator reasoning under the stated independent factors.

supports Source: A stochastic variational framework for fitting and diagnosing generalized linear mixed models · Locator: Section 5.2 prior-message construction and Jensen argument, PDF p. 19

Mechanism #likelihood-message

At the same local state, let G_i and F_i be expected first and second cumulant derivatives under the Gaussian predictor. For positive-exposure Poisson observations, both equal E_i*exp(t_i+v_i/2), elementwise, where t_i=V_i*m+Z_i*m_i and v_i=diag(V_i*Q*V_i^T+Z_i*Q_i*Z_i^T). For Bernoulli-logit they are standard-normal expectations of sigmoid(T) and sigmoid(T)*(1-sigmoid(T)), with T=t_ij+sqrt(v_ij)*Z0 and Z0~N(0,1). Define R_i=Z_i^T*diag(F_i)*Z_i and s_i=Z_i^T*(y_i-G_i). If R_i is positive definite, the standalone likelihood message is N(m_lik,C_lik), with C_lik=R_i^-1 and m_lik=m_i+C_lik*s_i. Positive semidefinite curvature alone does not suffice: rank-deficient Z_i gives a singular R_i and no proper Gaussian density on the full r-dimensional space, even when the combined posterior factor is proper. The normalized expectations here follow the 2013 Appendix B; the 2014 Appendix B display on PDF p. 37 and the 2013 Section 4.3 display on PDF p. 8 both omit a half in the Gaussian exponent.

supports Source: A stochastic variational framework for fitting and diagnosing generalized linear mixed models · Locator: Section 5.2 likelihood-message expressions, PDF p. 20; Appendix B cumulant definitions, PDF pp. 37–38
supports Source: Variational Inference for Generalized Linear Mixed Models Using Partially Noncentered Parametrizations · Locator: Appendix B normalized Gaussian construction, PDF pp. 19–20; contrast Section 4.3 display, PDF p. 8

Mechanism #fixed-point-combination

At a local fixed point, Q_i^-1=A+R_i and A*(m_i-B_i*m)=s_i. Thus the prior and likelihood natural parameters add to those of q(a_i): Q_i^-1*m_i=A*m_prior+R_i*m_lik. Equivalently, the ratio q(a_i)/N(a_i;m_prior,C_prior) is proportional to the likelihood Gaussian when its precision is positive definite. Away from the fixed point, summing the current messages produces the local Gaussian proposal, not necessarily the stored q(a_i); the ratio interpretation and diagnostic must not silently assume that stale local parameters satisfy these equations.

supports Source: A stochastic variational framework for fitting and diagnosing generalized linear mixed models · Locator: Section 5.2 natural-parameter sum and ratio argument, PDF p. 20; local optimization requirement, PDF p. 21

Mechanism #gaussian-conflict-score

Under the additional approximation that the two message-based replicates are independent, their difference delta=a_prior-a_lik is Gaussian with mean d=m_prior-m_lik and covariance C=C_prior+C_lik. In one dimension define the lower tail P(delta<=0)=Phi(-d/sqrt(C)), the upper tail as one minus that value, and the two-sided score as twice their minimum. State the selected tail explicitly. For r-dimensional positive-definite C, the source uses d^T*C^-1*d and the upper chi-square tail with r degrees of freedom under a Gaussian-difference approximation. These are probabilities or reference-tail scores from the constructed approximation; this algebra alone does not establish uniform frequentist p-values or false-positive control under repeated data.

supports Source: A stochastic variational framework for fitting and diagnosing generalized linear mixed models · Locator: Section 5.1 difference and tail definitions, PDF p. 18; independent Gaussian-message approximation, PDF p. 19
supports Source: Identifying outliers in Bayesian hierarchical models: a simulation-based approach · Locator: Section 4, Equations 7–11, journal pp. 418–420 / PDF pp. 10–12; Appendix A1 calibration distinctions, journal pp. 429–430 / PDF pp. 21–22

Limitation #leave-one-out-boundary

The simulation-based comparator draws a predictive-prior replicate using hyperparameters fitted without the assessed unit and a likelihood replicate from that unit with a reference prior. The variational messages instead use the full-data fitted factors, replace hyperparameter mixtures by expected-log Gaussian messages, and neglect replicate dependence. The likelihood-message interpretation invokes a uniform reference prior, whereas the source comparison uses Jeffreys priors for its likelihood replications. Nuisance fixed effects learned from other units can still couple the two replications. Full-data fitting, factorization, reference-prior choice and approximate Gaussian likelihood messages therefore prevent treating this diagnostic as exact leave-one-unit-out conflict analysis. The source describes its approximation argument as nonrigorous.

supports Source: A stochastic variational framework for fitting and diagnosing generalized linear mixed models · Locator: Sections 5.1–5.2, PDF pp. 17–20; Appendix C reference-prior likelihood replications, PDF p. 39
supports Source: Identifying outliers in Bayesian hierarchical models: a simulation-based approach · Locator: Sections 3–4, journal pp. 415–420 / PDF pp. 7–12; Section 6 full-data approximations, journal pp. 422–424 / PDF pp. 14–16

Scope #screening-and-evaluation

The source proposes the message diagnostic as an initial screen, with selected units eligible for closer Monte Carlo assessment. Its comparisons use small GLMM datasets and transformed tail-probability agreement with simulation-based comparators; those comparisons do not establish universal calibration, multiple-testing control or a clinical decision rule. Changing the centered/partially noncentered factorization can change the approximate messages and scores. A common invertible linear transformation of an already constructed Gaussian difference preserves its Mahalanobis statistic, but this algebraic invariance does not imply invariance across separately fitted variational parametrizations. No diagnostic threshold is made intrinsic to this concept.

supports Source: A stochastic variational framework for fitting and diagnosing generalized linear mixed models · Locator: Section 5.2 screening and parametrization discussion, PDF pp. 19–21; Sections 6.1–6.2 and Appendix D diagnostic comparisons, PDF pp. 21–28 and 40–42

⚠️ Curated Limitations & Negative Results

The prior and likelihood messages need not equal genuine predictive-prior and likelihood-posterior distributions. Independence, full-data substitution and Gaussian approximation are substantive assumptions; large cluster counts alone do not prove their accuracy.
A standalone likelihood message requires positive-definite R_i. A proper combined q(a_i) does not repair an improper separated message, and a pseudoinverse silently changes the procedure. Numerical quadrature and near-singular curvature can affect both message parameters and tail scores.
Tail orientation is explicit because the original 2007 discussion contains a reversed displayed difference mean on journal p. 419 / PDF p. 11 and an inconsistent inequality in Figure 4 caption on journal p. 439 / PDF p. 31. This record follows delta=prior-likelihood and the corresponding probability algebra, not those conflicting display fragments.
The Gaussian reference tail is not a general null calibration theorem. Multiple tested units, estimated hyperparameters, reference-prior choice, data reuse and selected follow-up require separate assessment. No automated rejection threshold, FDR guarantee or domain diagnosis is asserted.
All 42 pages of the 2014 preprint, 36 pages of the 2007 journal PDF and 21 pages of the supporting 2013 reprint were read. No author software, model fitting or empirical calibration simulation was run. The independent calculation constructs one local Poisson stationary block under supplied globals, compares its message with an exact scalar likelihood posterior, and checks Gaussian and singular-message boundaries; it does not optimize the global factors or reproduce reported experiments.

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:RandomEffectsModel` Random Effects Model	related_match	The diagnostic assesses agreement between a clustered random-effects prior and unit-specific likelihood information in a fitted Bayesian GLMM. RandomEffectsModel supplies the defining model context; it does not denote this message-separation diagnostic. The association is method-to-model relatedness, without equivalence or is-a.

Evidence Sources & Primary Literature

3 Sources

Source ID	Publication / Document	Type / Access	Reference & Checksum
`tan2014`	A stochastic variational framework for fitting and diagnosing generalized linear mixed models Linda S. L. Tan, David J. Nott	primary paper full_text	ID: https://doi.org/10.1214/14-BA885 sha256:276ebdc67a5f…
`marshall2007`	Identifying outliers in Bayesian hierarchical models: a simulation-based approach E. C. Marshall, D. J. Spiegelhalter	primary paper full_text	ID: https://doi.org/10.1214/07-BA218 sha256:289815a344a2…
`tan2013`	Variational Inference for Generalized Linear Mixed Models Using Partially Noncentered Parametrizations Linda S. L. Tan, David J. Nott	primary paper full_text	ID: https://doi.org/10.1214/13-STS418 sha256:32f56daf18e0…

Curation Provenance

Curator: Codex

Method: ai_assisted

Model: GPT-6

Created: 2026-10-13T00:34:42Z

Updated: 2026-10-13T00:34:42Z

Source File: data/curated/GaussianMessageConflictDiagnostic.yaml