Statistical Foundations

FULL EXECUTABLE FLOW ALGEBRA — STATISTICAL FOUNDATIONS
Kireth Research — Semantic Library
URL: https://kirethresearch.org/booksai-core/ontologies/probability-statistics-ontologies/statistical-foundations.html

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

BEGIN_FLOW

StatisticalFoundations :=
    Data
    ∧ Population
    ∧ Sample
    ∧ Variables
    ∧ ProbabilityModel
    ∧ StatisticalInference
    ∧ UncertaintyQuantification
    ∧ EvidenceEvaluation

Population := CompleteSetOfRelevantUnitsOrOutcomes
StatisticalUnit := EntityOnWhichMeasurementsAreObserved
Sample := ObservationsObtainedFromPopulation
SampleSize := n
Observation := RecordedValueAssociatedWithStatisticalUnit
Dataset := Observations ∧ Variables ∧ Metadata
Variable := MeasurableOrRecordedCharacteristic

VariableType := {
    Categorical,
    Numerical,
    Discrete,
    Continuous,
    Binary,
    Ordinal,
    Nominal
}

Parameter := PopulationOrModelQuantity
Statistic := FunctionOfSampleData
Estimator := StatisticUsedToEstimateParameter
Estimate := RealisedValueOfEstimator

SamplingDistribution :=
    DistributionOfStatistic
    under repeated sampling model

ProbabilityModel :=
    MathematicalRepresentationOfUncertainty

RandomVariable := OutcomeToMeasurableValueMapping
Distribution := ProbabilityLawOfRandomVariable
ExpectedValue := E[X]
Variance := Var(X)
StandardDeviation := sqrt(Var(X))
Covariance := Cov(X,Y)
Correlation := StandardisedCovariance

DescriptiveStatistics := SummariesOfObservedData
InferentialStatistics := SampleToPopulationOrModelReasoning

SampleMean :=
    X_bar=(1/n)Σ_iX_i

SampleVariance :=
    S²=[1/(n-1)]Σ_i(X_i-X_bar)²

SampleStandardDeviation := sqrt(S²)
InterquartileRange := Q3-Q1

FrequencyDistribution :=
    ValuesOrCategories
    → CountsOrRelativeFrequencies

RelativeFrequency := Count/TotalCount

EmpiricalCDF :=
    F_n(x)=(1/n)Σ_i1_(X_i≤x)

DataGeneratingProcess := MechanismProducingObservations
SamplingDesign := ProcedureSelectingUnitsOrObservations
SimpleRandomSample := ProbabilitySamplingWithSpecifiedEqualSelectionStructure
IIDSample := Independent ∧ IdenticallyDistributed

StratifiedSampling :=
    Population
    → PartitionIntoStrata
    → SampleWithinStrata

ClusterSampling :=
    Population
    → PartitionIntoClusters
    → SelectClustersOrUnits

SystematicSampling :=
    RandomisedStart
    → RegularIntervalSelection

ConvenienceSample := AccessibilityBasedSampling

SamplingBias :=
    SamplingProcess
    systematically distorts population representation

SelectionBias := InclusionMechanismAffectsInference
NonresponseBias := ResponseMechanismCreatesSystematicDifference
MeasurementError := ObservedValueDiffersFromTargetQuantity
RandomError := UnpredictableErrorComponent
SystematicError := PersistentStructuredError

Bias :=
    E[Estimator]-Parameter

UnbiasedEstimator :=
    E[Estimator]=Parameter

Consistency :=
    Estimator
    converges_to TargetParameter
    under asymptotic regime

Efficiency := RelativePrecisionUnderSpecifiedCriterion

MeanSquaredError :=
    MSE
    =
    Variance(Estimator)
    +
    Bias(Estimator)^2

StandardError :=
    StandardDeviationOfSamplingDistribution
    or estimate thereof

PointEstimate := SingleValueEstimate
IntervalEstimate := RangeGeneratedByInferentialProcedure

ConfidenceInterval :=
    Procedure
    with specified repeated-sampling coverage
    under model assumptions

ConfidenceLevel := 1-α

CoverageProbability :=
    P(IntervalProcedureContainsTrueParameter)

Hypothesis := StatementAboutPopulationOrModelParameter
NullHypothesis := H0
AlternativeHypothesis := H1
TestStatistic := StatisticUsedToAssessEvidenceRelativeToH0
NullDistribution := DistributionOfTestStatisticUnderH0

PValue :=
    ProbabilityUnderH0
    of result at least as incompatible with H0
    as observed
    according_to specified test

SignificanceLevel := α
RejectRule := PValue≤α under specified procedure
TypeIError := RejectH0WhenH0True
TypeIIError := FailToRejectH0WhenSpecifiedAlternativeTrue
StatisticalPower := 1-β
EffectSize := QuantitativeMagnitudeOfDifferenceOrAssociation
PracticalSignificance := DomainSpecificImportanceOfEffect
StatisticalSignificance := InferentialResultRelativeToSignificanceCriterion

StatisticalSignificanceDoesNotImplyPracticalImportance :=
    StatisticalSignificance ↛ PracticalImportance

PValueDoesNotEqualProbabilityNullTrue :=
    PValue ≠ P(H0|Data)
    in ordinary frequentist testing

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

MaximumLikelihoodEstimator :=
    θ_hat=argmax_θ L(θ|Data)

BayesianInference :=
    Prior ∧ Likelihood → Posterior

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

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

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

CredibleInterval :=
    IntervalContainingSpecifiedPosteriorProbability

FrequentistInference :=
    SamplingDistribution
    ∧ EstimatorOrTest
    → LongRunOperatingProperties

Model :=
    StructuredRepresentation
    linking variables and probability assumptions

ParametricModel := FiniteDimensionalParameterModel
NonparametricModel := NonFixedFiniteDimensionalModel
SemiparametricModel := ParametricComponent ∧ NonparametricComponent
ModelAssumption := ConditionRequiredForModelInterpretationOrInference

IndependenceAssumption :=
    JointDistributionFactorisesAccordingToModel

NormalityAssumption :=
    SelectedVariableOrErrorFollowsNormalDistributionWhereRequired

Homoscedasticity := ConstantVarianceAcrossSpecifiedConditions
Heteroscedasticity := VarianceChangesAcrossSpecifiedConditions
LinearityAssumption := ExpectedResponseRelatedLinearlyToPredictors

ModelMisspecification :=
    AssumedModelDiffersMateriallyFromDataGeneratingProcess

Residual := ObservedValue-FittedValue
ResidualAnalysis := Residuals → DiagnoseAssumptionsAndFit
GoodnessOfFit := ModelCompatibilityWithObservedData
Overfitting := SampleSpecificNoiseCapturedReducingGeneralisation
Underfitting := RelevantStructureInsufficientlyCaptured

Resampling := {
    Bootstrap,
    PermutationTest,
    CrossValidation
}

Bootstrap :=
    ResampleObservedData
    → ApproximateSamplingDistribution

PermutationTest :=
    ReassignLabelsOrValues
    under null exchangeability
    → ReferenceDistribution

CrossValidation :=
    PartitionData
    → TrainAndEvaluate
    → EstimatePredictivePerformance

Randomisation :=
    RandomAssignment
    used_to support causal inference

ObservationalStudy := TreatmentOrExposureNotRandomlyAssigned
ExperimentalStudy := ControlledIntervention
Confounding := VariableAssociatedWithExposureAndOutcome
Association := StatisticalRelationship
Causation := CausalEffectRequiringCausalAssumptionsOrDesign

AssociationDoesNotImplyCausation :=
    Association ↛ Causation

MissingData := IntendedObservationUnavailable

MissingnessMechanism := {
    MCAR,
    MAR,
    MNAR
}

Outlier := ObservationUnusuallyDistantUnderSpecifiedReference
RobustStatistics := MethodsLessSensitiveToExtremeObservationsOrModelDeviation

Standardisation :=
    Z=(X-Mean)/StandardDeviation

LawOfLargeNumbers :=
    SampleAverage
    converges_to Expectation
    under theorem conditions

CentralLimitTheorem :=
    StandardisedAggregate
    converges_in_distribution
    toward Normal
    under theorem conditions

AsymptoticDistribution := LimitingDistributionOfEstimatorOrStatistic

Identifiability :=
    DistinctParameterValues
    imply distinct probability distributions
    under model

Sufficiency :=
    StatisticRetainsAllSampleInformationAboutParameter
    under formal criterion

SufficientStatistic := T(Data)
AncillaryStatistic := StatisticWhoseDistributionDoesNotDependOnParameter

FisherInformation :=
    ExpectedCurvatureOrScoreVariance
    measuring information about parameter

ScoreFunction :=
    U(θ)=∂/∂θ log L(θ|Data)

CramerRaoRelation :=
    VarianceLowerBoundForUnbiasedEstimators
    under regularity conditions

StatisticalDecisionTheory :=
    ParameterOrState
    ∧ Action
    ∧ Loss
    → DecisionRule

LossFunction := L(action,state)
RiskFunction := ExpectedLoss

StatisticalEvidence :=
    Data
    ∧ Model
    ∧ InferentialProcedure
    ∧ Assumptions
    ∧ Uncertainty

Reproducibility := ConsistentComputationalResultsFromSameDataAndMethods
Replicability := CompatibleEvidenceInNewDataOrStudy

StatisticalWorkflow :=
    DefineQuestion
    → DefinePopulationAndVariables
    → DesignSamplingOrExperiment
    → CollectData
    → ValidateData
    → ExploreData
    → SpecifyModel
    → EstimateParameters
    → QuantifyUncertainty
    → TestOrCompareModels
    → DiagnoseAssumptions
    → InterpretResults
    → PreserveProvenance

StatisticalState := {
    ResearchQuestion,
    Population,
    Sample,
    Variables,
    SamplingDesign,
    ProbabilityModel,
    Parameters,
    Statistics,
    Estimates,
    Uncertainty,
    Assumptions,
    Evidence,
    Confidence,
    Provenance
}

StatisticalConstraint :=
    ConstraintOn(
        SamplingDesign,
        VariableType,
        Model,
        Parameters,
        Assumptions,
        Inference
    )

StatisticalConsistency :=
    Data
    ∧ SamplingDesign
    ∧ Model
    ∧ InferentialProcedure
    ∧ Interpretation
    jointly_compatible

InvalidInference :=
    ViolationOfRequiredAssumptions
    ∨ IncorrectSamplingModel
    ∨ ParameterStatisticConfusion
    ∨ InvalidProbabilityInterpretation
    ∨ UnsupportedCausalClaim

StatisticalValidation :=
    ResolveQuestion
    → ResolvePopulation
    → ResolveSample
    → ResolveVariables
    → ResolveSamplingDesign
    → ResolveModel
    → ValidateAssumptions
    → ResolveEstimatorOrTest
    → ValidateInference
    → ValidateInterpretation

SEMANTIC_EQUIVALENCE(StatisticalModelA,StatisticalModelB) :=
    same(TargetEstimandOrParameter)
    ∧ same(ObservableProbabilityStructure)
    ∧ same(InferentialScope)
    under semantically equivalent assumptions

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

CoreClasses := {
    Population,
    StatisticalUnit,
    Sample,
    Observation,
    Dataset,
    Variable,
    Parameter,
    Statistic,
    Estimator,
    Estimate,
    SamplingDistribution,
    ProbabilityModel,
    DescriptiveStatistics,
    InferentialStatistics,
    ConfidenceInterval,
    HypothesisTest,
    Likelihood,
    BayesianInference,
    FrequentistInference,
    StatisticalModel,
    SamplingDesign,
    StatisticalEvidence
}

CoreRelations := {
    sampledFrom,
    observedOn,
    measures,
    estimates,
    hasParameter,
    hasStatistic,
    distributedAs,
    conditionedOn,
    inferredFrom,
    assumes,
    tests,
    supports,
    contradicts,
    predicts,
    dependsOn,
    equivalentTo,
    tracesTo
}

TypedRelations :=
    Sample --sampledFrom--> Population
    Observation --observedOn--> StatisticalUnit
    Variable --measures--> Characteristic
    Estimator --estimates--> Parameter
    StatisticalModel --hasParameter--> Parameter
    Dataset --hasStatistic--> Statistic
    Statistic --distributedAs--> SamplingDistribution
    PosteriorDistribution --conditionedOn--> Data
    Inference --inferredFrom--> Sample
    StatisticalModel --assumes--> ModelAssumption
    HypothesisTest --tests--> Hypothesis
    Evidence --supports--> StatisticalClaim
    StatisticalResult --tracesTo--> Provenance

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

    Unbiased(T,θ)
        ⇒ E[T]=θ,

    IIDSample(X_1,...,X_n)
        ∧ finite_mean
        ⇒
        SampleMean
        converges_to PopulationMean
        under LLN conditions,

    SuitableCLTConditions
        ⇒
        StandardisedSampleAggregate
        converges_in_distribution
        to Normal,

    PValue≤α
        ⇒
        RejectH0
        under specified testing procedure,

    StatisticalSignificance
        ↛ PracticalSignificance,

    Association
        ↛ Causation,

    InvalidRequiredAssumption
        ⇒ FlagInferenceRisk
}

CanonicalStatisticalRepresentation :=
    ParseStatisticalQuestion
    → ResolvePopulation
    → ResolveUnits
    → ResolveSample
    → ResolveVariables
    → ResolveSamplingDesign
    → ResolveProbabilityModel
    → ResolveParametersAndStatistics
    → ResolveInferenceFramework
    → ValidateAssumptions
    → Canonicalise

StatisticalFoundationsGraph :=
    G=(V,E)

V := {
    Populations,
    StatisticalUnits,
    Samples,
    Observations,
    Datasets,
    Variables,
    Parameters,
    Statistics,
    Estimators,
    SamplingDistributions,
    ProbabilityModels,
    Hypotheses,
    Tests,
    Intervals,
    Models,
    Assumptions,
    Evidence,
    ProvenanceNodes
}

E := {
    sampledFrom,
    observedOn,
    measures,
    estimates,
    hasParameter,
    hasStatistic,
    distributedAs,
    conditionedOn,
    inferredFrom,
    assumes,
    tests,
    supports,
    contradicts,
    predicts,
    dependsOn,
    equivalentTo,
    tracesTo
}

CONFIDENCE(StatisticalInferenceResult) :=
    weighted_mean(
        QuestionResolution,
        PopulationSampleResolution,
        SamplingDesignQuality,
        VariableResolution,
        ModelAdequacy,
        AssumptionValidity,
        InferentialValidity,
        ProvenanceQuality
    )

AI_QUERY_INTERPRETATION :=
    Query
    → DetectStatisticalIntent
    → IdentifyPopulationAndSample
    → IdentifyVariables
    → ResolveParameterOrEstimand
    → ResolveSamplingDesign
    → IdentifyRequestedSummaryInferenceOrPrediction

AI_REASONING_FLOW :=
    Query
    → ParseStatisticalQuestion
    → ResolvePopulation
    → ResolveSample
    → ResolveVariablesAndTypes
    → ResolveSamplingOrExperimentalDesign
    → ResolveProbabilityModel
    → ResolveParametersAndStatistics
    → SelectDescriptiveOrInferentialMethod
    → ValidateAssumptions
    → ComputeEstimateOrTestStatistic
    → QuantifyUncertainty
    → EvaluateEvidence
    → DiagnoseModel
    → DistinguishAssociationFromCausation
    → UpdateConfidence
    → PreserveProvenance
    → GenerateExplanation

AI_CONSUMPTION := {
    ParseStatisticalEntities,
    ParseTypedRelations,
    ResolvePopulations,
    ResolveSamples,
    ResolveVariables,
    ClassifyVariableTypes,
    ResolveSamplingDesigns,
    ResolveProbabilityModels,
    ResolveParameters,
    ResolveStatistics,
    ComputeDescriptiveStatistics,
    ResolveSamplingDistributions,
    ComputePointEstimates,
    ComputeIntervalEstimates,
    EvaluateHypothesisTests,
    EvaluateLikelihoods,
    ResolveBayesianUpdates,
    ValidateModelAssumptions,
    EvaluateBiasAndVariance,
    ResolveResamplingMethods,
    DistinguishAssociationFromCausation,
    ValidateStatisticalInference,
    CompareSemanticEquivalence,
    UpdateConfidence,
    PreserveProvenance,
    GenerateFormalExplanation
}

UPDATE_MODEL :=
    NewDataSamplingDesignModelOrEvidence
    → ValidateInput
    → IdentifyAffectedStatisticalClaims
    → UpdateStatisticalState
    → RecomputeStatistics
    → RecomputeEstimates
    → RecomputeUncertainty
    → ReevaluateAssumptions
    → RevalidateInference
    → RecomputeConfidence
    → PreserveProvenance
    → VersionModel

DataUpdate :=
    ChangedDataset
    ⇒ recompute(
        DescriptiveStatistics,
        Estimates,
        TestStatistics,
        ModelFit
    )

ModelUpdate :=
    ChangedProbabilityModel
    ⇒ reevaluate(
        Parameters,
        Likelihood,
        SamplingDistributions,
        Inference
    )

DesignUpdate :=
    ChangedSamplingOrExperimentalDesign
    ⇒ reevaluate(
        BiasRisk,
        Variance,
        Generalisability,
        CausalInterpretation
    )

FLOW_HASH_INPUT :=
    canonical(
        ONTOLOGY,
        CoreClasses,
        CoreRelations,
        InferenceRules,
        StatisticalFoundations,
        StatisticalWorkflow,
        StatisticalValidation,
        SEMANTIC_EQUIVALENCE,
        AI_REASONING_FLOW
    )

FLOW_HASH :=
    HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {
    StatisticalFoundationsEntityResolution,
    ResearchQuestion,
    Population,
    Sample,
    Variables,
    SamplingDesign,
    ProbabilityModel,
    Parameters,
    Statistics,
    Estimates,
    Uncertainty,
    HypothesisOrDecisionContext,
    AssumptionStatus,
    InferenceValidity,
    Confidence,
    Provenance,
    Explanation
}

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA