Home / Curated / Algorithms / Prior-to-posterior divergence conflict diagnostic

Prior-to-posterior divergence conflict diagnostic

Algorithms AIMech:PriorToPosteriorDivergenceConflictDiagnostic DRAFT

Aliases: Prior-to-posterior Rényi divergence prior-data conflict check, Relative belief prior-to-posterior divergence conflict diagnostic

A Bayesian prior-data conflict diagnostic algorithm that measures belief change from a proper prior g(theta) to the posterior g(theta | y) using the Rényi divergence R_alpha(y) of order alpha > 0 (including Kullback-Leibler divergence at alpha = 1 and log maximum relative belief as alpha -> infinity), calibrates the observed divergence R_alpha(y_obs) against its reference distribution under the prior predictive p(y), and extends to hierarchical priors g(theta_1 | theta_2)g(theta_2) by decomposing checking into a marginal prior-predictive divergence test for theta_2 and a posterior-integrated (or partial-posterior) conditional divergence test for theta_1 | theta_2.

Identity Notes & Concept Scope No normalized identity match for the prior-to-posterior divergence conflict diagnostic across all 442 pinned AIO records and 564 maintained IDs, names, aliases and scoped synonyms. Fixed-string searches across data, sources, research, history, and reviews returned only frontier-tracking references in prior discovery reports noting Nott, Wang, Evans, & Englert (2020) as a pending concept. Distinct from AIMech:PriorExpansionScoreConflictDiagnostic, which evaluates the local directional derivative (Fisher score) of a family of prior expansions around the baseline prior rather than the global prior-to-posterior Rényi divergence of a fixed prior. Distinct from AIMech:EvansMoshonovPriorDataConflictCheck (and sufficient-statistic prior predictive checks), which compare the prior predictive density p(t_obs) of a minimal sufficient statistic T rather than the sufficiency-invariant posterior-to-prior divergence R_alpha(y).

Claims & Grounded Evidence

4 Claims

Mechanism #renyi-relative-belief-discrepancy-invariance

The diagnostic defines the checking discrepancy as the prior-to-posterior Rényi divergence R_alpha(y) = (alpha - 1)^(-1) log int g(theta | y)^alpha g(theta)^(1 - alpha) dtheta = log ||RB(theta | y)||_{alpha - 1}, where RB(theta | y) = g(theta | y) / g(theta) is the relative belief ratio, and computes the tail probability p_alpha = P(R_alpha(Y) >= R_alpha(y_obs)) for Y ~ p(y) = int g(theta) p(y | theta) dtheta; because R_alpha(y) depends on the data only through the posterior distribution, it is automatically a function of any minimal sufficient statistic and invariant to its choice.

supports Source: Checking for Prior-Data Conflict Using Prior to Posterior Divergences · Locator: Section 2.1, Equations 2–4 and 6–7, lines 25–39
supports Source: Using Prior Expansions for Prior-Data Conflict Checking · Locator: Section 2.1, Equations 1–2, PDF pp. 4–5

Mechanism #hierarchical-conditional-and-marginal-divergence-decomposition

For a partitioned parameter theta = (theta_1, theta_2) with hierarchical prior g(theta_1 | theta_2) g(theta_2), the check decomposes into (i) a marginal divergence R_{alpha 2}(y) between g(theta_2 | y) and g(theta_2) calibrated under the joint prior predictive p(y), and (ii) an expected conditional divergence R_{alpha 1}(y) = E_{g(theta_2 | y_obs)}[R_alpha(theta_1 | y, theta_2)] calibrated under the mixed reference predictive distribution m(y) = int g(theta_2 | y_obs) p(y | theta) g(theta_1 | theta_2) dtheta_1 dtheta_2 (or its partial-posterior / leave-one-out variant).

supports Source: Checking for Prior-Data Conflict Using Prior to Posterior Divergences · Locator: Section 2.2, Equations 8–12, lines 41–55; Section 3.3, Example 3.3, lines 90–105

Mechanism #asymptotic-jeffreys-and-reference-prior-calibration

Under regularity conditions as sample size n -> infinity with true parameter theta*, R_alpha(y) - (d/2) log(n / (2*pi)) + (d / (2*alpha)) log(alpha) converges almost surely to -log g(theta*) + 0.5 log |I(theta*)|, where I(theta) is the per-observation Fisher information matrix, so p_alpha converges to P(g(theta)|I(theta)|^(-1/2) <= g(theta*)|I(theta*)|^(-1/2)) for theta ~ g(theta), yielding a limiting p-value of 1 under Jeffreys' prior (and under two-stage reference priors for the hierarchical checks).

supports Source: Checking for Prior-Data Conflict Using Prior to Posterior Divergences · Locator: Section 2.1, Equation 5, lines 33–34; Section 3.2, Equation 17, lines 75–85; Section 4, Equations 19–20, lines 121–131

Application #variational-and-gaussian-mixture-divergence-approximation

To avoid expensive Markov chain Monte Carlo posterior integration across M + 1 simulated prior-predictive datasets, the diagnostic replaces g(theta | y) with fast variational Bayes approximations q(theta) in the same exponential family as the prior (yielding closed-form Rényi divergences) or with Gaussian mixture variational posteriors combined with the closed-form Hershey-Olsen Kullback-Leibler divergence bound.

supports Source: Checking for Prior-Data Conflict Using Prior to Posterior Divergences · Locator: Section 5, Examples 5.1–5.2 and Equation 21, lines 133–155

⚠️ Curated Limitations & Negative Results

Because the asymptotic calibration P(g(theta)|I(theta)|^(-1/2) <= g(theta*)|I(theta*)|^(-1/2)) measures prior density relative to Jeffreys' prior |I(theta)|^(1/2) rather than Lebesgue measure, a flat proper prior on a bounded parameter (such as Beta(1, 1) on a Binomial proportion) can produce small divergence p-values near the boundary even when the prior predictive distribution is completely uniform.
In the hierarchical level-1 check, integrating over the full hyperparameter posterior g(theta_2 | y_obs) rather than the partial posterior g(theta_2 | y_obs \ R_{alpha 1}(y_obs)) uses the observed discrepancy information twice, inflating the reference predictive scale and making the check more conservative when level-1 conflict is strong.
Direct Monte Carlo evaluation requires computing posterior-to-prior divergences across M + 1 simulated datasets, and variational or Hershey-Olsen Gaussian mixture approximations introduce approximation error when the true posterior is poorly captured by the variational family.
All 749 lines of Nott, Wang, Evans, & Englert (2020, arXiv:1611.00113v1) and all 31 pages of Nott et al. (2020, arXiv:1902.10393v2) were read. No author variational Bayes or MCMC code for the Missouri cancer or Bristol Royal Infirmary datasets was executed; constructed numerical checks in research/discovery/prior-posterior-divergence-conflict-boundaries.py verify analytical Normal, Beta-Binomial, Normal-Inverse-Gamma, and Hershey-Olsen mixture boundaries only.

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:BayesianNetwork` Bayesian Network	related_match	The prior-to-posterior divergence conflict diagnostic checks prior-data consistency across single-level and hierarchical Bayesian models (directed acyclic graphs of parameters and data). AIO:BayesianNetwork supplies the probabilistic graphical modeling context rather than an algorithmic superclass.

Evidence Sources & Primary Literature

2 Sources

Source ID	Publication / Document	Type / Access	Reference & Checksum
`nott-wang2020`	Checking for Prior-Data Conflict Using Prior to Posterior Divergences David J. Nott, Xueou Wang, Michael Evans, Berthold-Georg Englert	primary paper full_text	ID: https://doi.org/10.1214/19-STS731 sha256:8347841fa2b6…
`nott-seah2020`	Using Prior Expansions for Prior-Data Conflict Checking David J. Nott, Maxine Seah, Luai Al-Labadi, Michael Evans, Hui Khoon Ng, Berthold-Georg Englert	primary paper full_text	ID: https://doi.org/10.1214/20-BA1204 sha256:2d36e3af352a…

Ontology & Grounding Topology

Open in interactive network graph →

Curation Provenance

Curator: Antigravity

Method: ai_assisted

Model: Gemini 3

Created: 2026-10-17T09:16:00Z

Updated: 2026-10-17T09:16:00Z

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

Explore across X-Mech

Related computational & biological Mechs: Embedding Map Network Graph ProteinTraitsMech (ESM-2 pLM) DUFMech (Domain pLM) TraitMech CultureMech All 14 X-Mech Repositories