{ "schemaVersion": "1.6", "slug": "essential-hull-false-confidence", "title": "An essential convex hull criterion for false confidence in location models", "shortTitle": "Which false hypotheses can a probability favour? A convex hull answer", "url": "https://evidencepress.org/releases/essential-hull-false-confidence/", "oneLine": "In location models with light-tailed errors, a hypothesis is calibrated at every threshold at a parameter exactly when that parameter is not strictly inside its essential convex hull, and a finite test warns of high confidence in a false hypothesis.", "abstract": "Operators compute a probability of collision, and with large uncertainty it is small even if a collision is certain. Balch, Martin and Ferson (2019) named this false confidence and proved that affected hypotheses exist; Martin (2024) asked which. This candidate answers exactly for flat-prior posteriors in location models with radially decreasing errors. For exponential-power errors with p > 1, a hypothesis is valid at every threshold at a parameter exactly when that parameter lies outside the interior of the closed convex hull of the essential support of the hypothesis. The equivalence holds for a radial density g with tail probability S if and only if liminf S(R)/g(R − b) = 0 for every b > 0; tails like exp(−r log r) satisfy this in dimension one and fail in every higher dimension. Only almost-halfspaces are calibrated from both sides, which corrects a conjecture of Martin (2024) in these models, since a ball is not affected, and only strictly monotone functions of one linear form have exact marginal posteriors. With log-concave errors, if a closed convex set contains the complement of the hypothesis and has posterior mass at most s < 1/2 when the observation equals the truth, the hypothesis receives posterior probability at least 1 − s with sampling probability at least one half. The sufficient direction is elementary and halfspace validity is in Martin (2023). All-threshold failure may occur only at low thresholds. The model assumes known shape and exact matching. AI-assisted, unrefereed candidate; no external peer review, independent reproduction or formal verification is claimed.", "datePublished": "2026-11-04", "dateModified": "2026-11-04", "version": "0.1.0-candidate", "doi": "10.5281/zenodo.23948952", "doiUrl": "https://doi.org/10.5281/zenodo.23948952", "conceptDoi": "10.5281/zenodo.23948951", "pdfUrl": "https://github.com/ipitchford/essential-hull-false-confidence/releases/download/v0.1.0-candidate/essential-hull-false-confidence-0.1.0-candidate.pdf", "altPdfUrl": null, "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23948952", "repoUrl": "https://github.com/ipitchford/essential-hull-false-confidence", "releaseUrl": "https://github.com/ipitchford/essential-hull-false-confidence/releases/tag/v0.1.0-candidate", "markdownUrl": "https://evidencepress.org/releases/essential-hull-false-confidence/index.md", "bibtexUrl": "https://evidencepress.org/releases/essential-hull-false-confidence/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/essential-hull-false-confidence.mp3", "imageUrl": "https://evidencepress.org/assets/og/essential-hull-false-confidence.png?v=2eecb1f223", "coverArtUrl": "https://evidencepress.org/assets/art/essential-hull-false-confidence.svg?v=ff6872e7a4", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/essential-hull-false-confidence.mp3", "name": "Plain-English audio briefing", "description": "Communication aid; not additional research evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/essential-hull-false-confidence.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed AI-assisted statistics candidate. All results have written proofs; no proof assistant was used. The sufficient direction is elementary, halfspace validity is in Martin (2023), and the Gaussian converse combines classical Laplace-transform growth with a tail comparison. Internal AI-assisted review only, including a separate-model adversarial review, a re-review and a proof audit. External specialist review, independent reproduction and formal verification have not been established." }, "assurance": [ { "dimension": "availability", "label": "Availability and archiving", "question": "Is the evidence package publicly retrievable from an archive under a persistent identifier?", "state": "passed", "evidenceUrl": "https://zenodo.org/records/23948952", "note": "The versioned research package and manuscript are publicly archived on GitHub and Zenodo. Provider asset parity was checked against frozen local files; availability does not establish correctness or peer review." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "All four scripts were rerun with Python -I -B, with and without -O, from fresh copies (numpy 2.3.5, scipy 1.17.0, matplotlib 3.10.8). The JSON and the PNG figure matched the retained files exactly; the figure PDF differs only in its embedded date. Three corruption controls are detected. GitHub CI reruns the scripts on Linux." }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "No unaffiliated rerun has been established." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed", "note": "No independent reimplementation has been established." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "No proof-assistant formalisation." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "AI-assisted internal and adversarial review is not external specialist review." }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed", "note": "No external editorial peer review." }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "partial", "note": "requirements.txt pins numpy 2.3.5, scipy 1.17.0 and matplotlib 3.10.8; not a locked container; last-digit float differences across platforms are expected." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted analysis and computation", "AI-assisted internal review in several roles (Claude Code subagents); adversarial review and audit of new proofs by a separate model (OpenAI Codex); clean replay; re-review", "Claude Code: publication packaging and release" ], "humanRole": "Publication authority only; no human research contribution inferred.", "reviewStatus": "Internal AI-assisted and adversarial review, not external peer review." }, "attributionBoundary": null, "problem": null, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Prepared from the final paper (November 2026, with DOI line, 13 pages), its check scripts and the frozen release archive. The versioned GitHub and Zenodo research archives were checked against the frozen release assets on 4 November 2026.", "grandfatheredAtSchemaVersion": null, "keywords": [ "false confidence", "location model", "convex hull", "essential support", "confidence distribution", "fiducial inference", "satellite conjunction analysis", "log-concavity" ], "keyResults": [ "Hull criterion (Theorem 1, proof): under the tail condition (T), a hypothesis is valid at every threshold at a parameter exactly when that parameter is outside the interior of its essential convex hull; (T) is also necessary.", "Exact tail condition (Lemma 0, proof): exponential-power errors satisfy (T) exactly for p > 1; tails like exp(−r log r) satisfy it in dimension one and fail it in every higher dimension.", "Two-sided classification (Theorem 4, Corollary 5, proof): four calibration regimes; only almost-halfspaces are calibrated from both sides, correcting a conjecture of Martin (2024) in these models.", "Exact marginalisation (Theorem 6, proof): only strictly monotone functions of one linear form have exact marginal posteriors at every parameter.", "High confidence (Theorem 8, proof and Table 2): with log-concave errors, Q_θ(C) ≤ s < 1/2 for a closed convex C containing the complement gives posterior probability at least 1 − s at least half the time; at σ/R = 10 a certain collision gives Pc ≤ 0.01 with probability one." ], "reviews": [], "evidencePackage": "Written proofs of all results in a 13-page paper; three deterministic scripts check worked numbers in floating point (numpy 2.3.5, scipy 1.17.0), with failures that stay active under Python -O; replay from fresh copies reproduced the retained JSON exactly, and three corruption controls are detected.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Balch, Martin and Ferson (2019). Satellite conjunction analysis and the false confidence theorem. Proceedings of the Royal Society A. Names false confidence and proves that affected hypotheses exist.", "url": "https://doi.org/10.1098/rspa.2018.0565", "doi": "10.1098/rspa.2018.0565" }, { "citation": "Carmichael and Williams (2018). An exposition of the false confidence theorem. Stat. An exposition of the false confidence theorem.", "url": "https://doi.org/10.1002/sta4.201", "doi": "10.1002/sta4.201" }, { "citation": "Casella and Berger (1987). Reconciling Bayesian and Frequentist Evidence in the One-Sided Testing Problem. JASA. One-sided p-values and posterior probabilities.", "url": "https://doi.org/10.1080/01621459.1987.10478396", "doi": "10.1080/01621459.1987.10478396" }, { "citation": "Cunen, Hjort and Schweder (2020). Confidence in confidence distributions!. Proceedings of the Royal Society A. Confidence distributions for interval statements.", "url": "https://doi.org/10.1098/rspa.2019.0781", "doi": "10.1098/rspa.2019.0781" }, { "citation": "Efron and Tibshirani (1998). The problem of regions. The Annals of Statistics. First-order confidence value and curvature effects.", "url": "https://doi.org/10.1214/aos/1024691353", "doi": "10.1214/aos/1024691353" }, { "citation": "Elkantassi and Davison (2022). Space Oddity? A Statistical Formulation of Conjunction Assessment. Journal of Guidance, Control, and Dynamics. Likelihood inference on the miss distance.", "url": "https://doi.org/10.2514/1.G006282", "doi": "10.2514/1.G006282" }, { "citation": "Fraser (2011). Is Bayes Posterior just Quick and Dirty Confidence?. Statistical Science. Curvature effects for non-linear functionals.", "url": "https://doi.org/10.1214/11-STS352", "doi": "10.1214/11-STS352" }, { "citation": "Martin (2019). False confidence, non-additive beliefs, and valid statistical inference. International Journal of Approximate Reasoning. Non-additive beliefs and validity.", "url": "https://doi.org/10.1016/j.ijar.2019.06.005", "doi": "10.1016/j.ijar.2019.06.005" }, { "citation": "Martin (2024). Which Statistical Hypotheses are Afflicted with False Confidence?. Lecture Notes in Computer Science. Asks which hypotheses are affected, with the conjecture corrected here.", "url": "https://doi.org/10.1007/978-3-031-67977-3_15", "doi": "10.1007/978-3-031-67977-3_15" }, { "citation": "Martin (2026). Possibilistic Inferential Models: A Review. JASA. A review that still treats the characterisation as open.", "url": "https://doi.org/10.1080/01621459.2025.2606127", "doi": "10.1080/01621459.2025.2606127" }, { "citation": "Saumard and Wellner (2014). Log-concavity and strong log-concavity: A review. Statistics Surveys. Log-concavity, used for Theorem 8.", "url": "https://doi.org/10.1214/14-SS107", "doi": "10.1214/14-SS107" }, { "citation": "Shimodaira (2004). Approximately unbiased tests of regions using multistep-multiscale bootstrap resampling. The Annals of Statistics. Multiscale bootstrap for regions.", "url": "https://doi.org/10.1214/009053604000000823", "doi": "10.1214/009053604000000823" }, { "citation": "Shimodaira (2008). Testing regions with nonsmooth boundaries via multiscale bootstrap. Journal of Statistical Planning and Inference. Nonsmooth boundaries.", "url": "https://doi.org/10.1016/j.jspi.2007.04.001", "doi": "10.1016/j.jspi.2007.04.001" }, { "citation": "Martin (2023). Fiducial inference viewed through a possibility-theoretic inferential model lens. PMLR 215. Contains halfspace validity as a special case of Theorem 5.", "url": "https://proceedings.mlr.press/v215/martin23a.html", "doi": null } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:essential-hull-false-confidence", "attemptIds": [ "ep-attempt:essential-hull-false-confidence-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "communication" ], "lineageId": null, "accelerationPrimitives": [ "adversarial-controls", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "other", "description": "Which hypotheses about a location parameter can be favoured by a flat-prior posterior when false: exactly those with the parameter strictly inside the essential convex hull, under the tail condition (T), with a finite-level warning for log-concave errors.", "scope": "Location models with radially decreasing errors, known shape and exact matching of sampling and posterior laws." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "The paper separates the elementary sufficient direction, halfspace validity in Martin (2023) and the classical Laplace-transform ingredient from the classification and its consequences, and distinguishes all-threshold validity from high-confidence failure.", "remainingRisks": [ "Dated searches are not exhaustive priority certification.", "Unknown scale, informative priors and approximate posteriors are not covered." ] }, "humanJudgmentGates": [ "Publication is authorised; scientific endorsement is not inferred.", "External specialist review and independent reproduction remain open." ], "parentLinks": [], "assuranceTarget": { "dimensions": [ "specialistReview", "independentRerun", "formalVerification", "priorityAssessment" ], "nextAction": "Have a specialist in inferential models or confidence distributions check Theorem 1(b),(c) and Theorem 8.", "claimCeiling": "Proved results in the stated location models. Internal verification is not external assurance." }, "impactClaims": [ { "id": "essential-hull-false-confidence-no-impact", "aim": "science", "outcome": "An openly inspectable answer to a published question on false confidence.", "setting": "AI-assisted statistics research", "status": "NO_IMPACT_EVIDENCE", "designClass": "none", "comparator": "None registered.", "estimand": "No research-speed or scientific-impact effect is estimated.", "evidenceRefs": [], "registeredDesignRef": null, "independentAssessment": null } ] }, "researchMetrics": { "schemaVersion": "1.0", "policyUrl": "https://evidencepress.org/api/research-metrics-policy.json", "outcomePolicyUrl": "https://evidencepress.org/api/v2/research-metrics-policy.json", "workLedgerUrl": "https://evidencepress.org/api/work-ledger.json", "claimCeiling": "These records describe scoped workflow inputs and outcomes. They support calibration and reference-class construction, but do not by themselves establish research acceleration, theorem correctness, novelty, independent assurance or impact.", "attempts": [ { "attemptId": "ep-attempt:essential-hull-false-confidence-publication", "status": "active", "resultClass": "pending", "metrics": { "schemaVersion": "1.0", "measurementScope": "publication-only", "scopeBoundary": "Only archive packaging, site integration, guarded delivery and canonical readback from registration are prospective. Earlier research, review, revision and extension work are left-censored.", "forecast": { "frozenAt": "2026-11-04T12:33:00Z", "procedureClass": "full-candidate-publication", "targetOutcome": "Accurately scoped GitHub, Zenodo and canonical Evidence Press release with explanatory media and machine readback, if all publication gates pass.", "expectedActiveMinutes": 90, "plausibleLowMinutes": 60, "plausibleHighMinutes": 150, "expectedUnattendedWaitMinutes": 30, "referenceClass": { "label": "Full mathematical candidate publication", "sampleSize": 0, "basis": "Route benchmark planning prior, not empirical calibration." }, "fermiComponents": [ { "component": "Research archive packaging, replay and identities", "count": 1, "lowMinutesPerUnit": 25, "centralMinutesPerUnit": 35, "highMinutesPerUnit": 60, "basis": "One frozen package with a rating-free public export, clean replay and DOI insertion." }, { "component": "Page, metadata and media integration", "count": 1, "lowMinutesPerUnit": 20, "centralMinutesPerUnit": 30, "highMinutesPerUnit": 45, "basis": "One reader-first release, drafted in parallel with verification." }, { "component": "Guarded deployment and canonical readback", "count": 1, "lowMinutesPerUnit": 15, "centralMinutesPerUnit": 25, "highMinutesPerUnit": 45, "basis": "Mandatory site tests, one merge cycle with ledger carry-forward, zero-cost deployment and public checks." } ], "probabilityPositiveSignal": 0.9, "probabilityTargetClosure": 0.85, "probabilityHorizonMinutes": 300, "assumptions": [ "The approved scientific package remains valid and configured providers remain available." ], "reforecastTriggers": [ "A publication gate reveals a scientific or integrity defect, or a provider outage blocks delivery." ], "stopRule": "Do not weaken assurance, integrity or cost gates to meet the forecast." }, "outcome": null }, "missingnessReason": null } ] } }