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Mixture-approximation node-splitting conflict diagnostic

Algorithms AIMech:MixtureApproximationNodeSplittingDiagnostic DRAFT

Aliases: Gaussian-mixture node-splitting conflict check, Mixture-approximated separator conflict diagnostic

A Bayesian directed acyclic graph (DAG) node-splitting conflict diagnostic algorithm that partitions observed data around a selected separator parameter block into parent-linked and child-linked subsets, approximates the parent-side and child-side split posterior distributions of the duplicated separator node using parametric Gaussian mixture densities while propagating shared parent-parameter uncertainty, forms the closed-form convolved Gaussian mixture density of the split-node difference, and evaluates the density-level tail probability at zero without nonparametric kernel density estimation.

Identity Notes & Concept Scope No normalized identity match for the mixture-approximation node-splitting conflict diagnostic across all 442 pinned AIO records and 563 maintained IDs, names, aliases and scoped synonyms. Fixed-string searches across data, sources, research, history, and reviews returned only references in the Round 168 discovery report and review exclusions noting Yang, Nott, & Presanis (2026) Section 3.2 (Algorithm 2) as a separate pending concept. Distinct from AIMech:SeparatorNodeConflictDiagnostic and AIMech:MultipleTestingNodeConflictDiagnostic, which define general separator node splitting and simultaneous multiple-testing adjustments using empirical MCMC difference samples, multivariate kernel density estimation, or single-Gaussian Mahalanobis approximations rather than closed-form Gaussian mixture convolution of split posteriors with shared parent-parameter propagation. Distinct from AIMech:SequentialScoreDiscrepancyConflictDiagnostic (Algorithm 1 of Yang et al. 2026), which evaluates conditional prior-expansion Fisher scores without duplicating the separator node or fitting mixture approximations to split posteriors.

Claims & Grounded Evidence

4 Claims

Mechanism #parent-child-split-with-shared-nuisance-propagation

For a selected separator parameter block theta_2 in a Bayesian directed acyclic graph with ancestor/parent parameter block theta_1 and partitioned data y = (y_pa, y_ch), the node-splitting construction duplicates theta_2 into parent-side copy theta_2^(1) informed by y_pa_obs and child-side copy theta_2^(2) informed by y_ch_obs under a reference prior p_R(theta_2^(2)), while propagating posterior uncertainty in shared parent parameters theta_1 from p(theta_1 | y_pa_obs) to the child submodel.

supports Source: Detecting Conflicts in Evidence Synthesis Models Using Score Discrepancies · Locator: Section 3.2, Equations 16–18 and Algorithm 2 Steps 1–2, PDF pp. 17–20
supports Source: Conflict Diagnostics in Directed Acyclic Graphs, with Applications in Bayesian Evidence Synthesis · Locator: Section 2.2, PDF pp. 4–5; Section 5.2, PDF p. 17

Mechanism #gaussian-mixture-split-posterior-approximation

Rather than applying multivariate kernel density estimation directly to Markov chain Monte Carlo difference draws, Algorithm 2 approximates the parent-side split posterior p(theta_2^(1) | y_pa_obs) and child-side split posterior p_ch(theta_2^(2) | y_ch_obs, y_pa_obs) by Gaussian mixture distributions sum_{k=1}^K w_k N(mu_k, Sigma_k) and sum_{j=1}^J v_j N(m_j, S_j) fitted to posterior draws (or formed directly from conditional Gaussian components).

supports Source: Detecting Conflicts in Evidence Synthesis Models Using Score Discrepancies · Locator: Section 3.2, Equations 16 and 19, Algorithm 2, PDF pp. 17–20; Appendix C, PDF pp. 40–42

Mechanism #closed-form-mixture-difference-convolution-tail

Under independent split copies approximated by Gaussian mixtures, the difference delta = theta_2^(1) - theta_2^(2) has the closed-form Gaussian mixture density p_hat_delta(delta) = sum_{k=1}^K sum_{j=1}^J w_k v_j N(delta; mu_k - m_j, Sigma_k + S_j), enabling Monte Carlo evaluation of the density-level conflict p-value P(p_hat_delta(delta) <= p_hat_delta(0)) using analytic mixture density evaluations at simulated draws and at zero.

supports Source: Detecting Conflicts in Evidence Synthesis Models Using Score Discrepancies · Locator: Section 3.2, Equations 19–20 and Algorithm 2 Step 3, PDF pp. 19–20; Appendix C, PDF pp. 40–42

Application #multimodal-and-non-gaussian-conflict-detection

Representing the split-posterior difference as a multi-component Gaussian mixture preserves skewness and multimodality that a single-Gaussian Mahalanobis approximation collapses, allowing the density-level tail test to detect conflict when zero falls in a low-density trough between mixture modes.

supports Source: Detecting Conflicts in Evidence Synthesis Models Using Score Discrepancies · Locator: Section 3.2, PDF pp. 19–21; Section 4, PDF pp. 21–28; Appendix C, PDF pp. 40–42

⚠️ Curated Limitations & Negative Results

Convolving marginal Gaussian mixtures as sum_{k,j} w_k v_j N(mu_k - m_j, Sigma_k + S_j) treats the parent-side and child-side copies theta_2^(1) and theta_2^(2) as marginally independent; if both copies depend strongly on a shared parent nuisance parameter theta_1 and are compared conditionally on paired draws of theta_1, ignoring between-copy covariance inflates the difference covariance by 2 * Cov(theta_2^(1), theta_2^(2)).
Requires choosing the number of mixture components (for example via BIC) and fitting component means and covariance matrices via expectation-maximization; poor component selection, singular component covariances, or heavy-tailed split posteriors can distort the estimated density level at zero.
Like general density-level node-splitting scores, the tail probability P(p_hat_delta(delta) <= p_hat_delta(0)) is not invariant to nonlinear reparameterizations of the separator node theta_2 and requires a proper child-side split posterior under the chosen reference prior p_R(theta_2^(2)).
All 52 pages of Yang, Nott, & Presanis (2026, arXiv:2511.02977v2) and 22 pages of Presanis et al. (2013, arXiv:1310.0628v1) were read. No author EM/MCMC implementation, epidemic model fit, or empirical repeated-data calibration benchmark was executed; constructed numerical checks in research/discovery/mixture-node-splitting-boundaries.py verify analytical mixture convolution and shared nuisance pairing boundaries only.

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:BayesianNetwork` Bayesian Network	related_match	The mixture-approximation node-splitting conflict diagnostic operates on a Bayesian directed acyclic graph (Bayesian network) by duplicating a separator node across parent and child subgraphs and approximating the split posteriors with Gaussian mixtures. AIO:BayesianNetwork supplies the graphical model representation context rather than an algorithmic superclass.

Evidence Sources & Primary Literature

2 Sources

Source ID	Publication / Document	Type / Access	Reference & Checksum
`yang2026`	Detecting Conflicts in Evidence Synthesis Models Using Score Discrepancies Fuming Yang, David J. Nott, Anne M. Presanis	primary paper full_text	ID: https://doi.org/10.48550/arXiv.2511.02977 sha256:4829e0d63d56…
`presanis2013`	Conflict Diagnostics in Directed Acyclic Graphs, with Applications in Bayesian Evidence Synthesis Anne M. Presanis, David Ohlssen, David J. Spiegelhalter, Daniela De Angelis	primary paper full_text	ID: https://doi.org/10.1214/13-STS426 sha256:9cff5092b47d…

Ontology & Grounding Topology

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Curation Provenance

Curator: Antigravity

Method: ai_assisted

Model: Gemini 3

Created: 2026-10-17T08:52:00Z

Updated: 2026-10-17T08:52:00Z

Source File: data/curated/MixtureApproximationNodeSplittingDiagnostic.yaml · Schema: curated_concept.yaml

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