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Prior-Expansion Marginal Likelihood Score Prior-Data Conflict Check

Evaluation Methods AIMech:PriorExpansionMarginalLikelihoodScorePriorDataConflictCheck DRAFT

Aliases: Prior-Expansion Score Conflict Check, Marginal Likelihood Score Prior-Data Conflict Diagnostic, Prior Expansion Score Statistic for Prior-Data Conflict

A Bayesian model criticism and prior-data conflict evaluation method that embeds a working prior p_0(theta) into a continuous expansion family p(theta | gamma) indexed by a hyperparameter gamma (with p(theta | gamma_0) = p_0(theta)) and evaluates the score of the log marginal likelihood S_gamma(y) = d/dgamma log p(y | gamma)|_{gamma_0} as a posterior expectation of the prior score, computing a prior-predictive tail probability on the variance-standardized score statistic to detect conflict between prior assumptions and observed data.

Identity Notes & Concept Scope No AIO class represents Bayesian prior-data conflict diagnostics; MachineLearningTask (AIO:MachineLearningTask) is the closest broad match covering model criticism and diagnostic evaluation tasks.

Claims & Grounded Evidence

4 Claims

Mechanism #posterior-expectation-identity-for-marginal-likelihood-score

For a smooth prior expansion p(theta | gamma) reducing to p_0(theta) at gamma = gamma_0, the gradient of the log marginal likelihood S_gamma(y) = grad_gamma log p(y | gamma)|_{gamma_0} equals the posterior expectation E[grad_gamma log p(theta | gamma)|_{gamma_0} | y, gamma_0], allowing the test statistic to be computed from baseline posterior MCMC samples without refitting expanded models.

supports Source: Using prior expansions for prior-data conflict checking · Locator: Section 1, Equation 1.1, p. 2; Section 2.1, Equation 2.1, p. 5

Mechanism #prior-predictive-quadratic-score-tail-calibration

Standardizing the score vector by its prior predictive covariance V_gamma = Cov_{Y ~ p(y|gamma_0)}(S_gamma(Y)) yields the quadratic discrepancy statistic Q(y) = S_gamma(y)^T V_gamma^{-1} S_gamma(y) and prior-predictive p-value p_S(y_obs) = P(Q(Y) >= Q(y_obs)), which recovers the Evans-Moshonov minimal sufficient statistic check in exponential families and connects locally to relative belief ratios.

supports Source: Using prior expansions for prior-data conflict checking · Locator: Section 1, Equations 1.2-1.3, p. 3; Section 2.2, Equation 2.4, p. 7; Section 3, Equations 3.1-3.2, pp. 9-11

Application #hierarchical-and-regression-prior-sensitivity-application

The prior-expansion score diagnostic is applied to detect componentwise prior-data conflict in Bayesian regression, generalized linear models, and hierarchical random-effects models using location, scale, and heavy-tailed Student-t prior expansions.

supports Source: Using prior expansions for prior-data conflict checking · Locator: Section 4, Examples 4.1-4.3, pp. 13-20

Limitation #local-first-order-expansion-direction-dependence-limitation

Because S_gamma(y) is a local first-order derivative evaluated at gamma = gamma_0, its sensitivity depends on the chosen parametric expansion family p(theta | gamma), and calibrating p_S(y_obs) requires simulating from the prior predictive distribution p(y | gamma_0) when analytical moments are unavailable.

supports Source: Using prior expansions for prior-data conflict checking · Locator: Section 2.1-2.2, pp. 5-8; Section 5 Discussion, pp. 20-21

⚠️ Curated Limitations & Negative Results

Because S_gamma(y) is a local first-order derivative evaluated at gamma = gamma_0, its sensitivity depends on the chosen parametric expansion family p(theta | gamma), and calibrating p_S(y_obs) requires simulating from the prior predictive distribution p(y | gamma_0) when analytical moments are unavailable.

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:MachineLearningTask` Machine Learning Task	broad_match	Prior-expansion marginal likelihood score checking is a diagnostic evaluation method for Bayesian statistical and machine learning models.

Evidence Sources & Primary Literature

3 Sources

Source ID	Publication / Document	Type / Access	Reference & Checksum
`r831-primary-pdf`	Using prior expansions for prior-data conflict checking David J. Nott, Micheal Seah, Luai Al-Labadi, Michael Evans, Hui Khoon Ng, Berthold-Georg Englert	primary paper full_text	ID: https://arxiv.org/abs/1902.10393 sha256:2d36e3af352a…
`r831-arxiv-abs`	Using prior expansions for prior-data conflict checking (arXiv abstract) David J. Nott, Micheal Seah, Luai Al-Labadi, Michael Evans, Hui Khoon Ng, Berthold-Georg Englert	primary paper abstract_only	ID: https://arxiv.org/abs/1902.10393 sha256:e5970500f821…
`r831-metadata-cr`	Using prior expansions for prior-data conflict checking (Crossref metadata) David J. Nott, Micheal Seah, Luai Al-Labadi, Michael Evans, Hui Khoon Ng, Berthold-Georg Englert	primary paper abstract_only	ID: https://arxiv.org/abs/1902.10393 sha256:fe7536d15e65…

Ontology & Grounding Topology

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Curation Provenance

Curator: Antigravity

Method: ai_assisted

Model: Gemini

Created: 2026-10-19T15:14:00Z

Updated: 2026-10-19T15:14:00Z

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

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