Interval Estimation Ontology

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

BEGIN_FLOW

IntervalEstimation := StatisticalEstimation ∧ LowerBound ∧ UpperBound ∧ CoverageFramework ∧ UncertaintyQuantification
Population := TargetCollectionOrGeneratingProcess
Sample := {X_1,...,X_n}
Parameter := θ
ParameterSpace := Θ
Estimand := PreciselyDefinedTargetQuantity
Estimator := θ_hat=T(X_1,...,X_n)
IntervalEstimator := [L(X),U(X)]
ObservedIntervalEstimate := [L(x),U(x)]
LowerBound := L(X)
UpperBound := U(X)
IntervalWidth := U(X)-L(X)
IntervalMidpoint := [L(X)+U(X)]/2

ConfidenceInterval := RandomInterval constructed_by procedure with specified frequentist coverage
CoverageProbability := P_θ(L(X)≤θ≤U(X))
ConfidenceLevel := 1-α
NominalCoverage := 1-α
ExactCoverage := CoverageProbability equals_or_meets nominal target according_to exact procedure
ApproximateCoverage := CoverageProbability approaches_or_approximates nominal target
ConfidenceIntervalInterpretation := Procedure has repeated-sampling coverage property
ConfidenceLevelNotPosteriorProbability := FrequentistConfidenceLevel ≠ PosteriorProbabilityOfParameter

OneSidedLowerInterval := [L(X),∞)
OneSidedUpperInterval := (-∞,U(X)]
TwoSidedInterval := [L(X),U(X)]
EqualTailedInterval := TailErrorAllocatedEqually
SymmetricInterval := θ_hat ± MarginOfError
MarginOfError := CriticalValue × StandardError
StandardError := SE(θ_hat)=sqrt(Var(θ_hat))
CriticalValue := QuantileOfReferenceDistribution
NormalCriticalValue := z_(1-α/2)
TCriticalValue := t_(1-α/2,ν)

WaldInterval := θ_hat ± z_(1-α/2)SE_hat(θ_hat)
WaldIntervalConstraint := Requires appropriate asymptotic normality and reliable SE

MeanKnownVarianceInterval :=
 X_bar ± z_(1-α/2)σ/sqrt(n)
 under applicable normal sampling model

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

DifferenceOfMeansInterval :=
 EstimateDifference ± CriticalValue×SE(Difference)

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

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

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

PopulationStandardDeviationInterval := SquareRootEndpoints(VarianceConfidenceInterval)

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

ProportionWaldCaution := CoverageMayBePoor for small samples or proportions near boundaries
WilsonScoreInterval := ScoreBasedIntervalForBinomialProportion
AgrestiCoullInterval := AdjustedWaldTypeIntervalForBinomialProportion
ClopperPearsonInterval := ExactBinomialIntervalBasedOnTestInversion

LikelihoodRatioInterval :=
 {θ:2[ℓ(θ_hat)-ℓ(θ)]≤CriticalThreshold}

ProfileLikelihoodInterval :=
 NuisanceParametersOptimisedOut
 → LikelihoodIntervalForTargetParameter

ScoreInterval := InvertScoreTest → ConfidenceSet
TestInversion := ParameterValuesNotRejectedByLevelAlphaTest → ConfidenceSet

PivotalQuantity := Q(X,θ) with known_or_parameter-free reference distribution
PivotIntervalConstruction := ProbabilityStatementForPivot → InvertForθ → ConfidenceInterval
StudentisedPivot := (Estimator-Parameter)/EstimatedStandardError

BootstrapInterval := ObservedSample → BootstrapReplicates → ApproximateInterval
BootstrapPercentileInterval := [Quantile_(α/2)(θ_hat*),Quantile_(1-α/2)(θ_hat*)]
BasicBootstrapInterval := [2θ_hat-Q_(1-α/2)(θ_hat*),2θ_hat-Q_(α/2)(θ_hat*)]
BootstrapTInterval := StudentisedBootstrapDistribution → Interval
BCaInterval := BiasCorrected ∧ Accelerated ∧ BootstrapInterval

BayesianIntervalEstimation := PosteriorDistribution → CredibleInterval
CredibleInterval := Interval C satisfying P(θ∈C|Data)=1-α
EqualTailedCredibleInterval := [PosteriorQuantile(α/2),PosteriorQuantile(1-α/2)]
HighestPosteriorDensityInterval := PosteriorRegion containing specified probability mass under density threshold
CredibleVsConfidenceInterval := CredibleInterval has posterior-probability interpretation whereas ConfidenceInterval has repeated-sampling coverage interpretation

PredictionInterval := IntervalForFutureObservation rather_than parameter
ToleranceInterval := Interval intended_to contain specified population proportion with stated confidence
SimultaneousConfidenceIntervals := MultipleIntervals with joint coverage control
BonferroniAdjustment := IndividualAlpha=α/m for m comparisons
ConfidenceBand := IntervalFunction over predictor_or_function domain
RegressionCoefficientInterval :=
 β_hat_j ± t_(1-α/2,ν)SE(β_hat_j)
 under applicable linear-model assumptions

MeanResponseInterval := IntervalForE[Y|X=x]
PredictionIntervalRegression := IntervalForNewYAtX=x
PredictionIntervalWidthRelation := PredictionInterval generally wider_than corresponding mean-response interval under same model_and_level

AsymptoticInterval := Estimator ∧ AsymptoticNormality → ApproximateConfidenceInterval
AsymptoticNormality := sqrt(n)(θ_hat-θ)→N(0,V(θ))
AsymptoticWaldInterval := θ_hat±z_(1-α/2)sqrt(V_hat(θ)/n)
DeltaMethodInterval := SmoothTransformation ∧ AsymptoticEstimatorDistribution → ApproximateIntervalForTransformedParameter

BoundaryParameter := Parameter constrained_to bounded space
BoundaryIntervalConstraint := NaiveSymmetricInterval may extend outside parameter space
ParameterSpaceRespect := IntervalEndpoints should respect admissible parameter domain or document approximation

IntervalPrecision := NarrowerInterval indicates greater precision only when level, estimand, method, and scale are comparable
ConfidenceLevelWidthTradeoff := HigherConfidenceLevel generally → WiderInterval under fixed data_and_method
SampleSizeWidthRelation := LargerSampleSize often → NarrowerInterval under regular conditions
VariabilityWidthRelation := GreaterSamplingVariability → WiderInterval under comparable method

MissingDataIntervalEstimation := MissingnessMethod ∧ PointEstimator ∧ VarianceEstimator → IntervalEstimate
MultipleImputationInterval := MultipleImputationEstimates ∧ WithinVariance ∧ BetweenVariance → CombinedInterval
SurveyIntervalEstimation := SamplingWeights ∧ Stratification ∧ Clustering ∧ DesignVariance → DesignAwareInterval
ClusterRobustInterval := Estimate ∧ ClusterRobustSE ∧ ReferenceDistribution → Interval
RobustInterval := RobustEstimator ∧ RobustVarianceEstimate → Interval

ModelMisspecificationRisk := IncorrectModel may produce invalid nominal coverage
CoverageError := ActualCoverage-NominalCoverage
Undercoverage := ActualCoverage<NominalCoverage
Overcoverage := ActualCoverage>NominalCoverage
IntervalCalibration := AssessCoverage across repeated samples_or_simulations
SimulationCoverageStudy := GenerateRepeatedDatasets → ConstructIntervals → MeasureCoverageFrequency

IntervalConstraint :=
 ConstraintOn(
  TargetParameter,
  ParameterSpace,
  SamplingDesign,
  StatisticalModel,
  Estimator,
  VarianceEstimator,
  ConfidenceLevel,
  ReferenceDistribution
 )

IntervalConsistency :=
 Estimand ∧ PointEstimator ∧ StandardError ∧ CriticalValue
 ∧ ConfidenceLevel ∧ SamplingFramework
 jointly_compatible

InvalidIntervalEstimation :=
 UndefinedTarget
 ∨ InvalidConfidenceLevel
 ∨ InvalidVarianceEstimate
 ∨ IncompatibleReferenceDistribution
 ∨ IgnoredDependence
 ∨ IgnoredSamplingDesign
 ∨ ViolatedCriticalAssumption
 ∨ ParameterSpaceViolation

IntervalEstimationValidation :=
 ResolveTargetParameter
 → ResolveParameterSpace
 → ResolveSampleAndData
 → ResolveStatisticalModel
 → ResolvePointEstimator
 → ResolveSamplingDistribution
 → ResolveVarianceAndStandardError
 → ResolveConfidenceLevel
 → SelectIntervalMethod
 → ComputeCriticalValuesOrPosteriorQuantiles
 → ComputeBounds
 → ValidateParameterSpace
 → ValidateCoverageFramework
 → ValidateResult

IntervalEstimationState := {
 Population,Sample,Data,Parameter,ParameterSpace,Estimand,StatisticalModel,
 PointEstimator,PointEstimate,IntervalMethod,ConfidenceOrCredibleLevel,
 StandardError,CriticalValue,LowerBound,UpperBound,IntervalWidth,
 CoverageFramework,Assumptions,Evidence,Confidence,Provenance
}

SEMANTIC_EQUIVALENCE(IntervalA,IntervalB) :=
 same(TargetParameterOrEstimand)
 ∧ same(DataScope)
 ∧ same(IntervalFramework)
 ∧ same(ConfidenceOrCredibleLevel)
 ∧ same(EstimatorDefinition)
 ∧ same(VarianceOrPosteriorMethod)

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

CoreClasses := {
 IntervalEstimation,IntervalEstimator,ConfidenceInterval,CredibleInterval,
 PredictionInterval,ToleranceInterval,LowerBound,UpperBound,ConfidenceLevel,
 CoverageProbability,MarginOfError,CriticalValue,StandardError,WaldInterval,
 TInterval,LikelihoodInterval,BootstrapInterval,BayesianInterval,
 StatisticalModel,Sample,Population
}

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

TypedRelations :=
 IntervalEstimator --estimates--> Parameter
 IntervalEstimator --targets--> Estimand
 IntervalEstimate --computedFrom--> Sample
 IntervalEstimate --hasLowerBound--> LowerBound
 IntervalEstimate --hasUpperBound--> UpperBound
 ConfidenceInterval --hasCoverage--> CoverageProbability
 ConfidenceInterval --hasLevel--> ConfidenceLevel
 WaldInterval --usesEstimator--> PointEstimator
 WaldInterval --usesStandardError--> StandardError
 WaldInterval --usesCriticalValue--> CriticalValue
 BootstrapInterval --derivedFrom--> BootstrapDistribution
 CredibleInterval --conditionedOn--> PosteriorDistribution
 IntervalEstimate --tracesTo--> Provenance

InferenceRules := {
 ValidLowerUpperBounds ⇒ IntervalWidth=UpperBound-LowerBound,
 ConfidenceLevel=1-α ⇒ TwoSidedEqualTailProbability=α/2,
 AsymptoticallyNormalEstimator ∧ ValidSE ⇒ CandidateWaldInterval=θ_hat±z_(1-α/2)SE,
 NormalIIDSample ∧ UnknownVariance ⇒ MeanIntervalUsesTReference,
 PosteriorDistribution ∧ CredibleLevel ⇒ CandidateCredibleIntervalFromPosterior,
 InvalidVarianceEstimate ⇒ FlagIntervalInvalid,
 IgnoredDependence ⇒ FlagCoverageRisk
}
CanonicalIntervalRepresentation :=
 ParseIntervalEstimationRequest
 → ResolvePopulation
 → ResolveSampleAndData
 → ResolveTargetParameterOrEstimand
 → ResolveParameterSpace
 → ResolveStatisticalModel
 → ResolvePointEstimator
 → ResolveSamplingOrPosteriorDistribution
 → ResolveStandardErrorOrPosteriorQuantiles
 → ResolveIntervalFrameworkAndLevel
 → SelectIntervalMethod
 → ComputeLowerAndUpperBounds
 → ValidateCoverageOrPosteriorMass
 → ValidateParameterSpace
 → PreserveProvenance

IntervalEstimationGraph := G=(V,E)

V := {
 IntervalEstimations,IntervalEstimators,ConfidenceIntervals,CredibleIntervals,
 PredictionIntervals,ToleranceIntervals,Parameters,Estimands,PointEstimators,
 PointEstimates,StandardErrors,CriticalValues,LowerBounds,UpperBounds,
 CoverageProbabilities,StatisticalModels,Samples,Populations,Evidence,ProvenanceNodes
}

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

CONFIDENCE(IntervalEstimationResolution) :=
 weighted_mean(
  TargetResolution,
  PopulationSampleResolution,
  ModelResolution,
  EstimatorResolution,
  VarianceOrPosteriorResolution,
  IntervalMethodValidity,
  CoverageFrameworkValidity,
  ProvenanceQuality
 )

AI_QUERY_INTERPRETATION :=
 Query
 → DetectIntervalEstimationIntent
 → IdentifyPopulationSampleAndTarget
 → IdentifyConfidenceCrediblePredictionOrToleranceInterval
 → ResolveConfidenceOrCredibleLevel
 → ResolveEstimatorVariancePosteriorAndReferenceDistribution

AI_REASONING_FLOW :=
 Query
 → ParseIntervalEstimationProblem
 → ResolvePopulation
 → ResolveSampleAndData
 → ResolveTargetParameterOrEstimand
 → ResolveParameterSpace
 → ResolveStatisticalModel
 → ResolvePointEstimator
 → ResolveSamplingDistributionOrPosterior
 → ResolveStandardErrorOrPosteriorQuantiles
 → ResolveIntervalLevelAndMethod
 → ComputeCriticalValues
 → ComputeLowerAndUpperBounds
 → ValidateParameterSpaceAndCoverageFramework
 → EvaluateCoverageRisk
 → UpdateConfidence
 → PreserveProvenance
 → GenerateExplanation

AI_CONSUMPTION := {
 ParseIntervalEstimationEntities,
 ParseTypedRelations,
 ResolvePopulations,
 ResolveSamples,
 ResolveParameters,
 ResolveEstimands,
 ResolveParameterSpaces,
 ResolveStatisticalModels,
 ResolvePointEstimators,
 ResolveSamplingDistributions,
 ResolvePosteriorDistributions,
 ComputeStandardErrors,
 ResolveConfidenceLevels,
 ResolveCriticalValues,
 ComputeWaldIntervals,
 ComputeTIntervals,
 ComputeProportionIntervals,
 ComputeVarianceIntervals,
 ComputeLikelihoodIntervals,
 ComputeProfileLikelihoodIntervals,
 ComputeBootstrapIntervals,
 ComputeCredibleIntervals,
 ComputePredictionIntervals,
 ComputeToleranceIntervals,
 ResolveSimultaneousIntervals,
 EvaluateCoverage,
 ValidateParameterSpace,
 ValidateIntervalEstimation,
 CompareSemanticEquivalence,
 UpdateConfidence,
 PreserveProvenance,
 GenerateFormalExplanation
}

UPDATE_MODEL :=
 NewDataModelEstimatorLevelOrEvidence
 → ValidateInput
 → IdentifyAffectedIntervalClaims
 → UpdateIntervalEstimationState
 → RecomputePointEstimateIfRequired
 → RecomputeStandardErrorOrPosterior
 → RecomputeCriticalValues
 → RecomputeIntervalBounds
 → ReevaluateCoverageAndParameterSpace
 → RecomputeConfidence
 → PreserveProvenance
 → VersionModel

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

LevelUpdate :=
 ChangedConfidenceOrCredibleLevel
 ⇒ recompute(
  CriticalValueOrPosteriorQuantiles,
  LowerBound,
  UpperBound,
  IntervalWidth
 )

MethodUpdate :=
 ChangedIntervalMethod
 ⇒ reevaluate(
  Assumptions,
  Bounds,
  CoverageFramework,
  SemanticEquivalence
 )

FLOW_HASH_INPUT :=
 canonical(
  ONTOLOGY,
  CoreClasses,
  CoreRelations,
  InferenceRules,
  IntervalEstimation,
  IntervalEstimationValidation,
  IntervalConstraint,
  SEMANTIC_EQUIVALENCE,
  AI_REASONING_FLOW
 )

FLOW_HASH := HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {
 IntervalEstimationEntityResolution,
 Population,
 Sample,
 StatisticalModel,
 TargetParameter,
 Estimand,
 ParameterSpace,
 PointEstimator,
 PointEstimate,
 IntervalFramework,
 IntervalMethod,
 ConfidenceOrCredibleLevel,
 StandardError,
 CriticalValue,
 LowerBound,
 UpperBound,
 IntervalWidth,
 CoverageProbability,
 CoverageRisk,
 ParameterSpaceStatus,
 ValidationStatus,
 Confidence,
 Provenance,
 Explanation
}

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA