Confidence Interval Ontology

FULL EXECUTABLE FLOW ALGEBRA — CONFIDENCE INTERVAL ONTOLOGY
Kireth Research — Semantic Library
URL: https://kirethresearch.org/booksai-core/ontologies/probability-statistics-ontologies/statistical-estimation/confidence-interval-ontology-2.html
FLOW_VERSION := "4.0"
FLOW_TYPE := "Full Executable Flow Algebra"
DOMAIN := "Probability & Statistics Ontologies"
SUBDOMAIN := "Statistical Estimation"
ONTOLOGY := "Confidence Interval Ontology"
PRIMARY_ENTITY := ConfidenceInterval
AUTHORSHIP := "Mark Higgins — Sky (Recursive AI)"
SOURCE_SITE := "https://kirethresearch.org"

BEGIN_FLOW

ConfidenceInterval := IntervalEstimator ∧ FrequentistCoverageProcedure ∧ LowerBound ∧ UpperBound ∧ ConfidenceLevel
Population := TargetCollectionOrGeneratingProcess
Sample := {X_1,...,X_n}
Parameter := θ
ParameterSpace := Θ
Estimand := PreciselyDefinedTargetQuantity
PointEstimator := θ_hat=T(X)
ConfidenceIntervalProcedure := [L(X),U(X)]
ObservedConfidenceInterval := [L(x),U(x)]
ConfidenceLevel := 1-α
CoverageProbability := P_θ(L(X)≤θ≤U(X))
NominalCoverage := 1-α
CoverageError := CoverageProbability-NominalCoverage
Undercoverage := CoverageProbability<NominalCoverage
Overcoverage := CoverageProbability>NominalCoverage

RepeatedSamplingInterpretation :=
 RepeatDataGeneration → ConstructIntervalEachSample → LongRunCoverage≈ConfidenceLevel

ConfidenceLevelNotPosteriorProbability :=
 FrequentistConfidenceLevel ≠ PosteriorProbabilityOfParameterGivenObservedData

IntervalWidth := U-L
IntervalMidpoint := (L+U)/2
MarginOfError := CriticalValue×StandardError
StandardError := SE(θ_hat)=sqrt(Var(θ_hat))
CriticalValue := QuantileOfReferenceDistribution

TwoSidedCI := [L,U]
OneSidedLowerCI := [L,∞)
OneSidedUpperCI := (-∞,U]
EqualTailedCI := LowerTailError=α/2 ∧ UpperTailError=α/2
NormalCriticalValue := z_(1-α/2)
TCriticalValue := t_(1-α/2,ν)

WaldCI := θ_hat±z_(1-α/2)SE_hat(θ_hat)
WaldValidity := ApproximateNormality ∧ ReliableStandardError ∧ CompatibleParameterSpace

MeanKnownVarianceCI :=
 X_bar±z_(1-α/2)σ/sqrt(n)

MeanUnknownVarianceCI :=
 X_bar±t_(1-α/2,n-1)S/sqrt(n)
 under iid normal sampling

DifferenceOfMeansCI :=
 EstimateDifference±CriticalValue×SE(Difference)

WelchCI :=
 (X_bar1-X_bar2)
 ±t_(1-α/2,ν_W)sqrt(S1²/n1+S2²/n2)

PairedMeanCI :=
 D_bar±t_(1-α/2,n-1)S_D/sqrt(n)

VarianceCI :=
 NormalIIDSample
 ⇒ [
 (n-1)S²/χ²_(1-α/2,n-1),
 (n-1)S²/χ²_(α/2,n-1)
 ]

StandardDeviationCI := SquareRootEndpoints(VarianceCI)

ProportionWaldCI :=
 p_hat±z_(1-α/2)sqrt(p_hat(1-p_hat)/n)

ProportionWaldRisk :=
 SmallN ∨ BoundaryProportion
 ⇒ PotentialPoorCoverage

WilsonScoreCI := ScoreBasedBinomialProportionInterval
AgrestiCoullCI := AdjustedWaldBinomialProportionInterval
ClopperPearsonCI := ExactBinomialTestInversionInterval

DifferenceOfProportionsCI :=
 (p_hat1-p_hat2)±CriticalValue×EstimatedStandardError

LogOddsRatioCI :=
 log(OR_hat)±z_(1-α/2)SE(logOR_hat)

OddsRatioCI := exp(LogOddsRatioCIEndpoints)

PivotalQuantity :=
 Q(X,θ)
 with known parameter-free reference distribution

PivotConstruction :=
 ProbabilityStatementForPivot → InvertForθ → ConfidenceInterval

StudentisedPivot := (θ_hat-θ)/SE_hat(θ_hat)

TestInversion :=
 ConfidenceSet={θ0:LevelAlphaTestDoesNotReject(θ0)}

LikelihoodRatioCI :=
 {θ:2[ℓ(θ_hat)-ℓ(θ)]≤CriticalThreshold}
 under applicable asymptotic conditions

ProfileLikelihoodCI :=
 ProfileNuisanceParameters → LikelihoodRatioConfidenceSet

ScoreCI := InvertScoreTest → ConfidenceInterval

BootstrapCI :=
 ObservedSample → BootstrapReplicates → BootstrapSamplingApproximation → Interval

BootstrapPercentileCI :=
 [Q*_(α/2),Q*_(1-α/2)]

BasicBootstrapCI :=
 [2θ_hat-Q*_(1-α/2),2θ_hat-Q*_(α/2)]

BootstrapTCI :=
 BootstrapStudentisedStatistic → Quantiles → ConfidenceInterval

BCaCI :=
 BiasCorrection ∧ AccelerationCorrection → AdjustedBootstrapQuantiles

BootstrapValidityConstraint :=
 ResamplingScheme must reflect data-generating dependence structure

AsymptoticNormality :=
 sqrt(n)(θ_hat-θ)→N(0,V(θ))

AsymptoticCI :=
 θ_hat±z_(1-α/2)sqrt(V_hat(θ)/n)

DeltaMethodCI :=
 SmoothFunction ∧ AsymptoticEstimator → CIForTransformedParameter

MonotonicTransformationCI :=
 CI_θ=[L,U] ∧ IncreasingMonotonic(g)
 ⇒ CI_gθ=[g(L),g(U)]
 when confidence-set transformation is valid

LogScaleCI :=
 ConstructCIOnLogScale → ExponentiateEndpoints

RegressionCoefficientCI :=
 β_hat_j±t_(1-α/2,ν)SE(β_hat_j)
 under applicable linear-model assumptions

MeanResponseCI := ConfidenceIntervalForE[Y|X=x]
PredictionIntervalDistinction := MeanResponseCI ≠ PredictionIntervalForFutureObservation

SimultaneousCI :=
 MultipleParameterIntervals ∧ JointCoverageControl

FamilyWiseCoverage :=
 P(AllIntervalsCoverTargets)

BonferroniCI :=
 IndividualAlpha=α/m
 for m intervals

ConfidenceBand := FunctionValuedIntervalOverDomain
PointwiseConfidenceBand := CoverageAtEachFixedPoint
SimultaneousConfidenceBand := JointCoverageAcrossSpecifiedDomain
ConfidenceRegion := MultidimensionalGeneralisationOfConfidenceInterval

ParameterBoundary := θ∈Boundary(Θ)
BoundaryRisk := SymmetricApproximation may violate parameter constraints
ParameterSpaceRespect := ConfidenceSet⊆Θ or approximation limitation documented

ConfidenceLevelWidthTradeoff :=
 HigherConfidenceLevel generally → WiderInterval

SampleSizeWidthRelation :=
 LargerN generally → NarrowerInterval under regular conditions

StandardErrorWidthRelation :=
 LargerStandardError → WiderInterval

ConfidenceIntervalPrecision :=
 Width indicates precision only under comparable level,target,scale,and method

FinitePopulationCorrection :=
 SamplingWithoutReplacement ∧ FinitePopulation
 → AdjustStandardError → AdjustConfidenceInterval

SurveyCI :=
 DesignBasedEstimate ∧ DesignBasedVariance → ConfidenceInterval

ClusterRobustCI :=
 Estimate ∧ ClusterRobustSE ∧ ReferenceDistribution → ConfidenceInterval

HeteroscedasticityRobustCI :=
 RegressionEstimate ∧ RobustCovariance → ConfidenceInterval

MissingDataCI :=
 MissingDataMethod ∧ Estimate ∧ CorrectVariancePropagation → ConfidenceInterval

MultipleImputationCI :=
 PooledEstimate ∧ WithinImputationVariance ∧ BetweenImputationVariance
 → CombinedStandardError → ConfidenceInterval

ConfidenceIntervalCalibration :=
 CompareActualCoverageWithNominalCoverage

SimulationCoverage :=
 Repeat(GenerateData,ConstructCI,CheckCoverage)
 → EstimatedCoverage

CoverageEstimate :=
 CoveredReplications/TotalReplications

IntervalMethodComparison :=
 Compare(Coverage,Width,BiasSensitivity,BoundaryBehaviour,ComputationalCost)

ConfidenceIntervalConstraint :=
 ConstraintOn(
 TargetParameter,
 ParameterSpace,
 StatisticalModel,
 SamplingDesign,
 PointEstimator,
 StandardError,
 ReferenceDistribution,
 ConfidenceLevel
 )

ConfidenceIntervalConsistency :=
 Estimand ∧ Estimator ∧ StandardError ∧ CriticalValue
 ∧ ConfidenceLevel ∧ SamplingFramework
 jointly_compatible

InvalidConfidenceInterval :=
 UndefinedTarget
 ∨ InvalidConfidenceLevel
 ∨ InvalidStandardError
 ∨ IncorrectReferenceDistribution
 ∨ IgnoredDependence
 ∨ IgnoredSamplingDesign
 ∨ CriticalAssumptionViolation
 ∨ UnresolvedParameterBoundary

ConfidenceIntervalValidation :=
 ResolveTargetParameter
 → ResolveParameterSpace
 → ResolveSampleAndData
 → ResolveStatisticalModel
 → ResolvePointEstimator
 → ResolveSamplingDistribution
 → ResolveVarianceAndStandardError
 → ResolveConfidenceLevel
 → SelectConfidenceIntervalMethod
 → ComputeCriticalValueOrInversionRule
 → ComputeBounds
 → ValidateParameterSpace
 → AssessCoverageProperties
 → ValidateResult

ConfidenceIntervalState := {
 Population,Sample,Parameter,ParameterSpace,Estimand,StatisticalModel,
 PointEstimator,PointEstimate,StandardError,ConfidenceLevel,Alpha,CriticalValue,
 IntervalMethod,LowerBound,UpperBound,IntervalWidth,CoverageProbability,
 CoverageError,Assumptions,Evidence,Confidence,Provenance
}

SEMANTIC_EQUIVALENCE(CIA,CIB) :=
 same(TargetParameterOrEstimand)
 ∧ same(DataScope)
 ∧ same(ConfidenceLevel)
 ∧ same(IntervalMethod)
 ∧ same(EstimatorDefinition)
 ∧ same(VarianceAndSamplingFramework)

ConfidenceIntervalOntology :=
 Classes ∧ Relations ∧ Constraints ∧ Axioms ∧ InferenceRules

CoreClasses := {
 ConfidenceInterval,ConfidenceIntervalProcedure,LowerConfidenceBound,UpperConfidenceBound,
 ConfidenceLevel,CoverageProbability,CoverageError,MarginOfError,StandardError,CriticalValue,
 WaldConfidenceInterval,TConfidenceInterval,ScoreConfidenceInterval,LikelihoodRatioConfidenceInterval,
 BootstrapConfidenceInterval,SimultaneousConfidenceInterval,ConfidenceBand,ConfidenceRegion,
 Parameter,Estimator,Sample,Population
}

CoreRelations := {
 estimates,targets,computedFrom,hasLowerBound,hasUpperBound,hasLevel,hasCoverage,
 hasCoverageError,usesEstimator,usesStandardError,usesCriticalValue,derivedFrom,
 adjustedFor,constrainedBy,equivalentTo,tracesTo
}

TypedRelations :=
 ConfidenceInterval --estimates--> Parameter
 ConfidenceInterval --targets--> Estimand
 ConfidenceInterval --computedFrom--> Sample
 ConfidenceInterval --hasLowerBound--> LowerConfidenceBound
 ConfidenceInterval --hasUpperBound--> UpperConfidenceBound
 ConfidenceInterval --hasLevel--> ConfidenceLevel
 ConfidenceInterval --hasCoverage--> CoverageProbability
 ConfidenceInterval --hasCoverageError--> CoverageError
 ConfidenceInterval --usesEstimator--> PointEstimator
 ConfidenceInterval --usesStandardError--> StandardError
 ConfidenceInterval --usesCriticalValue--> CriticalValue
 BootstrapConfidenceInterval --derivedFrom--> BootstrapDistribution
 SurveyConfidenceInterval --adjustedFor--> SamplingDesign
 ConfidenceInterval --tracesTo--> Provenance

InferenceRules := {
 ConfidenceLevel=1-α ⇒ Alpha=α,
 EqualTailedTwoSidedCI ⇒ TailProbability=α/2,
 ValidBounds ⇒ IntervalWidth=UpperBound-LowerBound,
 AsymptoticallyNormalEstimator ∧ ValidStandardError
   ⇒ CandidateWaldCI=θ_hat±z_(1-α/2)SE,
 NormalIIDSample ∧ UnknownVariance
   ⇒ MeanCIUsesStudentT,
 LevelAlphaTestFamily
   ⇒ ConfidenceSetByTestInversion,
 InvalidStandardError
   ⇒ FlagConfidenceIntervalInvalid,
 IgnoredDependence
   ⇒ FlagCoverageRisk
}

CanonicalConfidenceIntervalRepresentation :=
 ParseConfidenceIntervalRequest
 → ResolvePopulation
 → ResolveSampleAndData
 → ResolveTargetParameterOrEstimand
 → ResolveParameterSpace
 → ResolveStatisticalModel
 → ResolvePointEstimator
 → ResolveSamplingDistribution
 → ResolveVarianceAndStandardError
 → ResolveConfidenceLevel
 → SelectIntervalConstruction
 → ComputeCriticalValueOrTestInversion
 → ComputeLowerAndUpperBounds
 → AssessCoverage
 → ValidateParameterSpace
 → ValidateConfidenceInterval

ConfidenceIntervalGraph := G=(V,E)

V := {
 ConfidenceIntervals,ConfidenceIntervalProcedures,LowerBounds,UpperBounds,ConfidenceLevels,
 CoverageProbabilities,CoverageErrors,PointEstimators,PointEstimates,StandardErrors,
 CriticalValues,WaldIntervals,TIntervals,ScoreIntervals,LikelihoodIntervals,
 BootstrapIntervals,SimultaneousIntervals,ConfidenceBands,ConfidenceRegions,
 Parameters,Samples,Populations,Evidence,ProvenanceNodes
}

E := {
 estimates,targets,computedFrom,hasLowerBound,hasUpperBound,hasLevel,hasCoverage,
 hasCoverageError,usesEstimator,usesStandardError,usesCriticalValue,derivedFrom,
 adjustedFor,constrainedBy,equivalentTo,tracesTo
}

CONFIDENCE(ConfidenceIntervalResolution) :=
 weighted_mean(
 TargetResolution,
 PopulationSampleResolution,
 ModelResolution,
 EstimatorResolution,
 StandardErrorValidity,
 IntervalMethodValidity,
 CoverageAssessmentQuality,
 ProvenanceQuality
 )

AI_QUERY_INTERPRETATION :=
 Query
 → DetectConfidenceIntervalIntent
 → IdentifyPopulationSampleAndTarget
 → ResolveConfidenceLevel
 → IdentifyMeanProportionVarianceRegressionOrGeneralParameter
 → ResolveEstimatorStandardErrorReferenceDistributionAndIntervalMethod

AI_REASONING_FLOW :=
 Query
 → ParseConfidenceIntervalProblem
 → ResolvePopulation
 → ResolveSampleAndData
 → ResolveTargetParameterOrEstimand
 → ResolveParameterSpace
 → ResolveStatisticalModel
 → ResolvePointEstimator
 → ResolveSamplingDistribution
 → ComputeOrResolveStandardError
 → ResolveConfidenceLevelAndAlpha
 → SelectConfidenceIntervalMethod
 → ComputeCriticalValueOrInversionRule
 → ComputeLowerAndUpperBounds
 → ValidateParameterSpace
 → AssessCoverageAndWidth
 → UpdateConfidence
 → PreserveProvenance
 → GenerateExplanation

AI_CONSUMPTION := {
 ParseConfidenceIntervalEntities,
 ParseTypedRelations,
 ResolvePopulations,
 ResolveSamples,
 ResolveParameters,
 ResolveEstimands,
 ResolveParameterSpaces,
 ResolveStatisticalModels,
 ResolvePointEstimators,
 ResolveSamplingDistributions,
 ResolveStandardErrors,
 ResolveConfidenceLevels,
 ResolveCriticalValues,
 ComputeMeanConfidenceIntervals,
 ComputeDifferenceOfMeansIntervals,
 ComputeVarianceIntervals,
 ComputeStandardDeviationIntervals,
 ComputeProportionIntervals,
 ComputeDifferenceOfProportionsIntervals,
 ComputeOddsRatioIntervals,
 ComputeLikelihoodRatioIntervals,
 ComputeProfileLikelihoodIntervals,
 ComputeScoreIntervals,
 ComputeBootstrapIntervals,
 ComputeRegressionCoefficientIntervals,
 ComputeSimultaneousIntervals,
 EvaluateCoverage,
 ValidateParameterSpace,
 ValidateConfidenceIntervalSemantics,
 CompareSemanticEquivalence,
 UpdateConfidence,
 PreserveProvenance,
 GenerateFormalExplanation
}

UPDATE_MODEL :=
 NewDataModelEstimatorLevelOrEvidence
 → ValidateInput
 → IdentifyAffectedConfidenceIntervalClaims
 → UpdateConfidenceIntervalState
 → RecomputePointEstimateIfRequired
 → RecomputeStandardError
 → RecomputeCriticalValue
 → RecomputeLowerAndUpperBounds
 → ReevaluateCoverageAndParameterSpace
 → RecomputeConfidence
 → PreserveProvenance
 → VersionModel

DataUpdate :=
 ChangedSampleData
 ⇒ recompute(
 PointEstimate,
 StandardError,
 LowerBound,
 UpperBound,
 IntervalWidth
 )

LevelUpdate :=
 ChangedConfidenceLevel
 ⇒ recompute(
 Alpha,
 CriticalValue,
 LowerBound,
 UpperBound,
 IntervalWidth
 )

MethodUpdate :=
 ChangedConfidenceIntervalMethod
 ⇒ reevaluate(
 Assumptions,
 Bounds,
 CoverageProperties,
 SemanticEquivalence
 )

FLOW_HASH_INPUT :=
 canonical(
 ONTOLOGY,
 CoreClasses,
 CoreRelations,
 InferenceRules,
 ConfidenceInterval,
 ConfidenceIntervalValidation,
 ConfidenceIntervalConstraint,
 SEMANTIC_EQUIVALENCE,
 AI_REASONING_FLOW
 )

FLOW_HASH := HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {
 ConfidenceIntervalEntityResolution,
 Population,
 Sample,
 StatisticalModel,
 TargetParameter,
 Estimand,
 ParameterSpace,
 PointEstimator,
 PointEstimate,
 StandardError,
 ConfidenceLevel,
 Alpha,
 CriticalValue,
 IntervalMethod,
 LowerBound,
 UpperBound,
 IntervalWidth,
 CoverageProbability,
 CoverageError,
 ParameterSpaceStatus,
 ValidationStatus,
 Confidence,
 Provenance,
 Explanation
}

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA