Inferential Statistics Ontology

FULL EXECUTABLE FLOW ALGEBRA — INFERENTIAL STATISTICS ONTOLOGY
Kireth Research — Semantic Library
URL: https://kirethresearch.org/booksai-core/ontologies/probability-statistics-ontologies/statistical-foundations/inferential-statistics-ontology.htm

FLOW_VERSION := "4.0"
FLOW_TYPE := "Full Executable Flow Algebra"
DOMAIN := "Probability & Statistics Ontologies"
SUBDOMAIN := "Statistical Foundations"
ONTOLOGY := "Inferential Statistics Ontology"
PRIMARY_ENTITY := InferentialStatistics
AUTHORSHIP := "Mark Higgins — Sky (Recursive AI)"
SOURCE_SITE := "https://kirethresearch.org"

BEGIN_FLOW

InferentialStatistics :=
    SampleEvidence
    ∧ PopulationOrModelTarget
    ∧ ProbabilityModel
    ∧ Estimation
    ∧ HypothesisTesting
    ∧ UncertaintyQuantification
    ∧ ModelAssessment

Population := CompleteSetOfRelevantUnitsOrOutcomes
Sample := ObservedSubsetOrSequenceFromPopulation
SampleSize := n
Parameter := UnknownPopulationOrModelQuantity
Statistic := FunctionOfSampleData
Estimator := StatisticUsedToEstimateParameter
Estimate := RealisedEstimatorValue
Estimand := PreciselyDefinedTargetQuantity

InferenceTarget := {
    Parameter,
    Estimand,
    Prediction,
    Hypothesis,
    Model,
    CausalEffect
}

ProbabilityModel :=
    DistributionalStructure
    linking Parameters to possible Data

SamplingDistribution :=
    DistributionOfStatistic
    under repeated sampling

PointEstimation :=
    Sample → Estimator → PointEstimate

EstimatorBias :=
    Bias(T)=E[T]-θ

UnbiasedEstimator :=
    E[T]=θ

Consistency :=
    T_n → θ
    under specified asymptotic conditions

MeanSquaredError :=
    MSE(T)
    =
    Var(T)+Bias(T)^2

StandardError :=
    SD(SamplingDistribution)
    or estimate thereof

IntervalEstimation :=
    Data → LowerBound ∧ UpperBound

ConfidenceInterval :=
    RandomIntervalProcedure
    with specified coverage probability

ConfidenceLevel := 1-α

CoverageProbability :=
    P_θ(L(Data)≤θ≤U(Data))

Hypothesis := StatementAboutParameterOrDistribution
NullHypothesis := H0
AlternativeHypothesis := H1
TestStatistic := T(Data)
NullDistribution := DistributionOfTestStatisticUnderH0
SignificanceLevel := α

TypeIError := RejectH0WhenH0True
TypeIIError := FailToRejectH0UnderSpecifiedAlternative
Power := 1-β

PValue :=
    ProbabilityUnderH0
    of statistic at least as incompatible
    with H0 as observed
    according_to test definition

DecisionRule :=
    PValue≤α
    ⇒ RejectH0
    under specified procedure

FailureToReject :=
    PValue>α
    ⇒ FailToRejectH0

FailureToRejectDoesNotProveNull :=
    FailToRejectH0 ↛ ProveH0True

EffectSize := MagnitudeOfDifferenceOrAssociation
PracticalSignificance := DomainSpecificImportance
StatisticalSignificance := ResultMeetingSignificanceCriterion

StatisticalSignificanceDoesNotImplyPracticalImportance :=
    StatisticalSignificance ↛ PracticalSignificance

Likelihood := L(θ|Data)
LogLikelihood := logL(θ|Data)

MaximumLikelihoodEstimator :=
    θ_hat=argmax_θL(θ|Data)

ScoreFunction :=
    U(θ)=∂/∂θ logL(θ|Data)

FisherInformation :=
    ExpectedScoreVarianceOrLikelihoodCurvature
    under regularity conditions

LikelihoodRatio :=
    SupLikelihoodUnderH0
    /
    SupLikelihoodUnderFullModel

LikelihoodRatioStatistic :=
    -2log(LikelihoodRatio)

WilksRelation :=
    LikelihoodRatioStatistic
    → ChiSquaredDistribution
    asymptotically
    under regularity conditions

WaldStatistic :=
    EstimateMinusNull
    /
    EstimatedStandardError

ScoreTest :=
    ScoreAtNull
    scaled_by InformationAtNull

AsymptoticInference :=
    LargeSampleApproximation
    using limiting distributions

LawOfLargeNumbers :=
    SampleAverage
    → ExpectedValue
    under theorem conditions

CentralLimitTheorem :=
    StandardisedAggregate
    → NormalDistribution
    under theorem conditions

AsymptoticNormality :=
    sqrt(n)(Estimator-Parameter)
    → NormalDistribution
    under method-specific conditions

DeltaMethod :=
    AsymptoticEstimatorDistribution
    → DistributionOfTransformation

Bootstrap :=
    ResampleObservedData
    → ApproximateSamplingDistribution

PermutationTest :=
    PermuteUnderNullExchangeability
    → ReferenceDistribution

ParametricInference :=
    SpecifiedDistributionFamily
    indexed_by parameters

NonparametricInference :=
    FewerOrDifferentDistributionalAssumptions

NuisanceParameter :=
    ParameterNotPrimaryTarget
    but affecting inference

SufficientStatistic :=
    StatisticRetainingAllInformationAboutParameter
    under formal criterion

PivotalQuantity :=
    FunctionOfDataAndParameter
    with parameter-free distribution

BayesianInference :=
    PriorDistribution
    ∧ Likelihood
    → PosteriorDistribution

PriorDistribution := p(θ)
PosteriorDistribution := p(θ|Data)

BayesTheorem :=
    p(θ|Data)
    ∝
    p(Data|θ)p(θ)

PosteriorMean := E[θ|Data]
MAPEstimate := argmax_θp(θ|Data)

CredibleInterval :=
    Interval C
    such_that
    P(θ∈C|Data)
    =
    specified posterior probability

PosteriorPredictive :=
    p(X_new|Data)
    =
    Integral p(X_new|θ)p(θ|Data)dθ

BayesFactor :=
    P(Data|M1)/P(Data|M2)

PosteriorOdds :=
    PriorOdds × BayesFactor

ModelSelection :=
    CandidateModels
    → CompareFitComplexityOrEvidence
    → SelectOrAverage

AIC :=
    2k-2logL_hat

BIC :=
    klogn-2logL_hat

CrossValidation :=
    PartitionData
    → FitOnTraining
    → EvaluateOnValidation
    → EstimatePredictivePerformance

MultipleTesting :=
    MultipleHypotheses
    → IncreasedFalsePositiveRisk

FamilyWiseErrorRate :=
    P(AtLeastOneTypeIError)

BonferroniCorrection :=
    PerTestAlpha=α_family/m

FalseDiscoveryRate :=
    ExpectedFalseDiscoveryProportion
    among rejected hypotheses
    under formal definition

RegressionInference :=
    RegressionModel
    → CoefficientEstimates
    → StandardErrors
    → TestsAndIntervals

MeanInference :=
    SampleMean
    → StandardError
    → NormalOrTReferenceDistribution

VarianceInference :=
    SampleVariance
    → ChiSquaredOrFDistribution
    under normality assumptions

ProportionInference :=
    BinomialOrBernoulliData
    → EstimateProbability
    → QuantifyUncertainty

Prediction :=
    Model
    ∧ EstimatedParameters
    → FutureOutcomeDistribution

PredictionInterval :=
    IntervalForFutureObservation

ExperimentalInference :=
    RandomAssignment
    supports causal comparison
    under design assumptions

ObservationalInference :=
    Requires stronger assumptions
    for causal interpretation

Confounding :=
    VariableOrStructure
    distorting exposure-outcome relation

AssociationDoesNotImplyCausation :=
    StatisticalAssociation ↛ CausalEffect

SensitivityAnalysis :=
    VaryAssumptionsOrParameters
    → EvaluateInferenceRobustness

ModelDiagnostics :=
    Residuals
    ∧ Influence
    ∧ GoodnessOfFit
    ∧ AssumptionChecks

InferentialAssumption := {
    SamplingAssumption,
    Independence,
    DistributionalAssumption,
    VarianceAssumption,
    Exchangeability,
    Identifiability,
    CorrectModelSpecification
}

InferenceValidity :=
    Method
    ∧ Design
    ∧ Assumptions
    ∧ Computation
    ∧ Interpretation
    jointly_compatible

InvalidInference :=
    UnsupportedGeneralisation
    ∨ InvalidReferenceDistribution
    ∨ CriticalAssumptionViolation
    ∨ MisinterpretedPValue
    ∨ UnjustifiedCausalClaim
    ∨ UnaccountedMultipleTesting

InferentialWorkflow :=
    DefineTarget
    → ResolvePopulationAndSample
    → ResolveStudyDesign
    → SpecifyProbabilityModel
    → IdentifyParameterOrHypothesis
    → SelectEstimatorOrTest
    → ValidateAssumptions
    → ComputeEstimateOrStatistic
    → QuantifyUncertainty
    → EvaluateEvidence
    → DiagnoseModel
    → PerformSensitivityAnalysisIfNeeded
    → InterpretWithinScope
    → PreserveProvenance

InferentialStatisticsState := {
    Population,
    Sample,
    StudyDesign,
    ProbabilityModel,
    Parameter,
    Estimator,
    Estimate,
    SamplingDistribution,
    StandardError,
    Interval,
    Hypothesis,
    TestStatistic,
    PValue,
    EffectSize,
    Assumptions,
    Evidence,
    Confidence,
    Provenance
}

InferentialValidation :=
    ResolveTarget
    → ResolveDesign
    → ResolveModel
    → ResolveEstimatorOrTest
    → ValidateSamplingDistribution
    → ValidateAssumptions
    → ValidateUncertainty
    → ValidateEvidenceInterpretation
    → ValidateScopeOfConclusion

SEMANTIC_EQUIVALENCE(InferenceA,InferenceB) :=
    same(TargetParameterOrEstimand)
    ∧ same(DataSemantics)
    ∧ same(SamplingOrPosteriorModel)
    ∧ same(InferentialScope)
    under semantically equivalent assumptions

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

CoreClasses := {
    InferentialStatistics,
    Population,
    Sample,
    Parameter,
    Estimand,
    Statistic,
    Estimator,
    Estimate,
    SamplingDistribution,
    StandardError,
    ConfidenceInterval,
    Hypothesis,
    TestStatistic,
    PValue,
    Power,
    Likelihood,
    MaximumLikelihoodEstimator,
    BayesianInference,
    PosteriorDistribution,
    PredictiveDistribution,
    StatisticalEvidence
}

CoreRelations := {
    sampledFrom,
    estimates,
    targets,
    hasSamplingDistribution,
    hasStandardError,
    tests,
    hasPValue,
    hasPower,
    conditionedOn,
    updatesTo,
    predicts,
    assumes,
    supports,
    contradicts,
    derivedFrom,
    dependsOn,
    equivalentTo,
    tracesTo
}

TypedRelations :=
    Sample --sampledFrom--> Population
    Estimator --estimates--> Parameter
    Inference --targets--> Estimand
    Statistic --hasSamplingDistribution--> SamplingDistribution
    Estimator --hasStandardError--> StandardError
    HypothesisTest --tests--> Hypothesis
    HypothesisTest --hasPValue--> PValue
    HypothesisTest --hasPower--> Power
    PosteriorDistribution --conditionedOn--> Data
    PriorDistribution --updatesTo--> PosteriorDistribution
    PredictiveModel --predicts--> FutureObservation
    InferentialMethod --assumes--> InferentialAssumption
    StatisticalEvidence --supports--> StatisticalClaim
    InferentialResult --tracesTo--> Provenance

InferenceRules := {
    Estimator(T,θ)
        ⇒ Estimate=T(Data),

    Unbiased(T,θ)
        ⇒ E[T]=θ,

    PValue≤α
        ⇒ RejectH0
        under specified procedure,

    PValue>α
        ⇒ FailToRejectH0,

    FailToRejectH0
        ↛ ProveH0True,

    StatisticalSignificance
        ↛ PracticalSignificance,

    Association
        ↛ Causation,

    Prior
        ∧ Likelihood
        ⇒ Posterior
        via BayesTheorem,

    CriticalAssumptionViolation
        ⇒ FlagInferenceRisk
}

CanonicalInferentialRepresentation :=
    ParseInferenceQuestion
    → ResolvePopulation
    → ResolveSample
    → ResolveStudyDesign
    → ResolveTargetParameterOrEstimand
    → ResolveProbabilityModel
    → ResolveEstimatorTestOrPosterior
    → ResolveUncertaintyMeasure
    → ValidateAssumptions
    → ValidateInterpretation
    → CanonicaliseInference

InferentialStatisticsGraph :=
    G=(V,E)

V := {
    Populations,
    Samples,
    StudyDesigns,
    Parameters,
    Estimands,
    Statistics,
    Estimators,
    Estimates,
    SamplingDistributions,
    StandardErrors,
    ConfidenceIntervals,
    Hypotheses,
    TestStatistics,
    PValues,
    PowerFunctions,
    Likelihoods,
    Priors,
    Posteriors,
    PredictiveDistributions,
    Assumptions,
    Evidence,
    ProvenanceNodes
}

E := {
    sampledFrom,
    estimates,
    targets,
    hasSamplingDistribution,
    hasStandardError,
    tests,
    hasPValue,
    hasPower,
    conditionedOn,
    updatesTo,
    predicts,
    assumes,
    supports,
    contradicts,
    derivedFrom,
    dependsOn,
    equivalentTo,
    tracesTo
}

CONFIDENCE(InferentialStatisticsResult) :=
    weighted_mean(
        TargetResolution,
        PopulationSampleResolution,
        StudyDesignQuality,
        ProbabilityModelAdequacy,
        EstimatorOrTestValidity,
        AssumptionValidity,
        UncertaintyQuality,
        ProvenanceQuality
    )

AI_QUERY_INTERPRETATION :=
    Query
    → DetectInferentialStatisticsIntent
    → IdentifyPopulationSampleAndDesign
    → IdentifyParameterEstimandHypothesisOrPrediction
    → ResolveProbabilityModel
    → IdentifyRequestedEstimateIntervalTestOrPosterior

AI_REASONING_FLOW :=
    Query
    → ParseInferentialProblem
    → ResolvePopulation
    → ResolveSample
    → ResolveStudyDesign
    → ResolveTargetParameterOrEstimand
    → ResolveProbabilityModel
    → SelectEstimatorTestOrBayesianProcedure
    → ValidateAssumptions
    → ComputeEstimateOrTestStatistic
    → ResolveSamplingOrPosteriorDistribution
    → QuantifyUncertainty
    → ComputePValueIntervalPosteriorOrPredictionAsRequired
    → EvaluateEffectSizeAndPracticalContext
    → DiagnoseModel
    → EvaluateSensitivityIfRequired
    → ValidateScopeOfConclusion
    → UpdateConfidence
    → PreserveProvenance
    → GenerateExplanation

AI_CONSUMPTION := {
    ParseInferentialStatisticsEntities,
    ParseTypedRelations,
    ResolvePopulations,
    ResolveSamples,
    ResolveStudyDesigns,
    ResolveParameters,
    ResolveEstimands,
    ResolveEstimators,
    ResolveSamplingDistributions,
    ComputeStandardErrors,
    ComputeConfidenceIntervals,
    EvaluateHypothesisTests,
    ComputePValues,
    EvaluateStatisticalPower,
    EvaluateLikelihoods,
    ResolveMaximumLikelihoodEstimates,
    ResolveBayesianPosteriors,
    ComputeCredibleIntervals,
    ComputePosteriorPredictions,
    EvaluateModelSelection,
    ResolveMultipleTestingAdjustments,
    ValidateInferentialAssumptions,
    ValidateInferenceScope,
    CompareSemanticEquivalence,
    UpdateConfidence,
    PreserveProvenance,
    GenerateFormalExplanation
}

UPDATE_MODEL :=
    NewDataDesignModelParameterOrEvidence
    → ValidateInput
    → IdentifyAffectedInferentialClaims
    → UpdateInferentialState
    → RecomputeEstimates
    → RecomputeSamplingOrPosteriorDistributions
    → RecomputeUncertainty
    → ReevaluateHypothesisTestsOrPredictions
    → RevalidateAssumptions
    → RecomputeConfidence
    → PreserveProvenance
    → VersionModel

DataUpdate :=
    ChangedSampleData
    ⇒ recompute(
        Estimates,
        StandardErrors,
        TestStatistics,
        PValues,
        Intervals,
        ModelFit
    )

DesignUpdate :=
    ChangedStudyDesign
    ⇒ reevaluate(
        SamplingValidity,
        BiasRisk,
        Generalisability,
        CausalInterpretation
    )

ModelUpdate :=
    ChangedProbabilityModel
    ⇒ reevaluate(
        Likelihood,
        SamplingDistribution,
        Posterior,
        Predictions,
        InferentialValidity
    )

FLOW_HASH_INPUT :=
    canonical(
        ONTOLOGY,
        CoreClasses,
        CoreRelations,
        InferenceRules,
        InferentialStatistics,
        InferentialWorkflow,
        InferentialValidation,
        SEMANTIC_EQUIVALENCE,
        AI_REASONING_FLOW
    )

FLOW_HASH :=
    HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {
    InferentialStatisticsEntityResolution,
    Population,
    Sample,
    StudyDesign,
    ProbabilityModel,
    TargetParameterOrEstimand,
    Estimator,
    Estimate,
    SamplingOrPosteriorDistribution,
    StandardError,
    ConfidenceOrCredibleInterval,
    Hypothesis,
    TestStatistic,
    PValue,
    Power,
    EffectSize,
    Prediction,
    AssumptionStatus,
    InferenceValidity,
    Confidence,
    Provenance,
    Explanation
}

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA