Student's t Distribution Ontology

FULL EXECUTABLE FLOW ALGEBRA — STUDENT'S T-DISTRIBUTION ONTOLOGY
Kireth Research — Semantic Library
URL: https://kirethresearch.org/booksai-core/ontologies/probability-statistics-ontologies/probability-distributions/students-t-distribution-ontology.html

FLOW_VERSION := "4.0"
FLOW_TYPE := "Full Executable Flow Algebra"
DOMAIN := "Probability & Statistics Ontologies"
SUBDOMAIN := "Probability Distributions"
ONTOLOGY := "Student's T-Distribution Ontology"
PRIMARY_ENTITY := StudentsTDistribution
AUTHORSHIP := "Mark Higgins — Sky (Recursive AI)"
SOURCE_SITE := "https://kirethresearch.org"

BEGIN_FLOW

StudentsTDistribution :=
    ContinuousProbabilityDistribution
    ∧ SymmetricHeavyTailedDistribution
    ∧ DegreesOfFreedomParameter
    ∧ NormalChiSquaredRatioConstruction
    ∧ SmallSampleInferenceDistribution

ProbabilitySpace := (Ω,F,P)

TRandomVariable :=
    T~t_ν

DegreesOfFreedom :=
    ν

ParameterConstraint :=
    ν>0

Support :=
    R

StandardNormalComponent :=
    Z~N(0,1)

ChiSquaredComponent :=
    V~χ²_ν

IndependenceCondition :=
    Z⟂V

TConstruction :=
    T
    =
    Z
    /
    sqrt(V/ν)

RatioDistribution :=
    Z~N(0,1)
    ∧ V~χ²_ν
    ∧ Z⟂V
    ⇒
    Z/sqrt(V/ν)
    ~
    t_ν

StudentTPDF :=
    f_T(t)
    =
    Γ((ν+1)/2)
    /
    [
        sqrt(νπ)
        Γ(ν/2)
    ]
    ×
    (1+t²/ν)^(-(ν+1)/2)

PDFNonnegativity :=
    f_T(t)>0
    for all t∈R

PDFNormalisation :=
    Integral_{-∞}^{∞}
    f_T(t)dt
    =
    1

TCDF :=
    F_T(t)
    =
    P(T≤t)

Symmetry :=
    f_T(-t)
    =
    f_T(t)

CDFSymmetry :=
    F_T(-t)
    =
    1-F_T(t)

Median :=
    0

Mode :=
    0

Mean :=
    0
    when ν>1

MeanUndefined :=
    0<ν≤1
    ⇒
    ordinary expectation undefined

Variance :=
    ν/(ν-2)
    when ν>2

InfiniteVariance :=
    1<ν≤2
    ⇒
    variance infinite

VarianceUndefined :=
    0<ν≤1
    ⇒
    ordinary variance undefined

Skewness :=
    0
    when defined

ExcessKurtosis :=
    6/(ν-4)
    when ν>4

HeavyTails :=
    StudentT
    has heavier tails
    than StandardNormal
    for finite ν

TailDecay :=
    f_T(t)
    asymptotically proportional_to
    |t|^(-(ν+1))

DegreesOfFreedomEffect :=
    Increasingν
    → thinner tails
    → closer_to StandardNormal

NormalLimit :=
    ν→∞
    ⇒
    t_ν
    → N(0,1)
    in distribution

CauchySpecialCase :=
    t_1
    ≡
    StandardCauchy

LocationScaleT :=
    X
    =
    μ+sT
    where
    T~t_ν
    ∧ s>0

LocationScaleTPDF :=
    f_X(x)
    =
    (1/s)
    f_T((x-μ)/s)

TQuantile :=
    q_p
    such_that
    P(T≤q_p)=p

CriticalValue :=
    t_(p,ν)

TwoSidedCriticalValue :=
    t_(1-α/2,ν)

OneSidedCriticalValue :=
    t_(1-α,ν)

NormalSample :=
    X_1,...,X_n
    iid N(μ,σ²)

SampleMean :=
    X_bar
    =
    (1/n)Σ_iX_i

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

NormalSampleMeanDistribution :=
    (X_bar-μ)/(σ/√n)
    ~
    N(0,1)

SampleVarianceChiSquared :=
    (n-1)S²/σ²
    ~
    χ²_(n-1)

NormalMeanVarianceIndependence :=
    X_bar
    ⟂
    S²

OneSampleTStatistic :=
    T
    =
    (X_bar-μ)
    /
    (S/√n)

OneSampleTDistribution :=
    Under normal iid assumptions:
        T~t_(n-1)

OneSampleDegreesOfFreedom :=
    ν=n-1

OneSampleMeanConfidenceInterval :=
    X_bar
    ±
    t_(1-α/2,n-1)
    S/√n

OneSampleTTest :=
    H0:μ=μ0
    → Compute
    T
    =
    (X_bar-μ0)/(S/√n)

TwoSampleIndependentT :=
    DifferenceOfSampleMeans
    standardised_by
    estimated standard error

PooledTTest :=
    IndependentNormalSamples
    ∧ EqualVarianceAssumption
    → PooledVarianceTStatistic

PooledDegreesOfFreedom :=
    n1+n2-2

WelchTTest :=
    IndependentSamples
    ∧ UnequalVarianceAllowed
    → WelchTStatistic

WelchStatistic :=
    T
    =
    (X_bar1-X_bar2)
    /
    sqrt(
        S1²/n1
        +
        S2²/n2
    )

WelchSatterthwaiteDF :=
    ν
    approximately
    =
    (S1²/n1+S2²/n2)²
    /
    [
        (S1²/n1)²/(n1-1)
        +
        (S2²/n2)²/(n2-1)
    ]

PairedTTest :=
    PairedObservations
    → Differences D_i
    → OneSampleTTestOnDifferences

PairedDegreesOfFreedom :=
    n_pairs-1

RegressionTRelation :=
    EstimatedRegressionCoefficient
    standardised_by
    its estimated standard error
    may follow t distribution
    under linear-model assumptions

RegressionTStatistic :=
    t
    =
    (β_hat_j-β_j0)
    /
    SE(β_hat_j)

RegressionDegreesOfFreedom :=
    ResidualDegreesOfFreedom

CorrelationTRelation :=
    Under bivariate-normal null assumptions:
    t
    =
    r
    sqrt(
        (n-2)/(1-r²)
    )
    ~
    t_(n-2)

BayesianTRelation :=
    StudentTDistribution
    arises as
    marginal_or_predictive distribution
    when Gaussian variance uncertainty
    is integrated out
    under conjugate models

NormalInverseGammaRelation :=
    GaussianLikelihood
    ∧ NormalInverseGammaPrior
    → StudentTPredictive

ScaleMixtureRepresentation :=
    StudentT
    =
    NormalDistribution
    with random inverse-gamma-related variance

GaussianScaleMixture :=
    T|τ
    ~
    N(0,1/τ)
    ∧ τ
    ~
    Gamma(ν/2,ν/2)
    ⇒
    T~t_ν

RobustModellingRelation :=
    HeavyTails
    make StudentT likelihood
    less sensitive to extreme observations
    than Gaussian likelihood
    in some models

StudentTLikelihood :=
    Data
    modelled_with
    location-scale StudentT distribution

LogLikelihood :=
    SumOfLogStudentTDensities
    over observations

FDistributionRelation :=
    T~t_ν
    ⇒
    T²
    ~
    F_(1,ν)

ChiSquaredRelation :=
    TConstruction
    directly depends_on
    χ²_ν denominator component

NormalRelation :=
    ν→∞
    ⇒
    StudentTApproachesNormal

TDistributionMomentExistence :=
    E[|T|^k]
    finite
    iff
    k<ν

OddMoments :=
    Symmetry
    ⇒
    existing odd moments
    equal 0

Entropy :=
    DefinedUsing
    BetaAndDigammaFunctions
    for ν>0

StudentTConstraint :=
    ConstraintOn(
        DegreesOfFreedom,
        Support,
        PDF,
        MomentExistence,
        InferentialAssumptions
    )

DegreesOfFreedomConstraint :=
    ν>0

SupportConstraint :=
    T∈R

NormalisationConstraint :=
    Integral f_T(t)dt=1

MomentConstraint :=
    RequestedMomentOrder
    must satisfy
    existence conditions

InferenceAssumptionConstraint :=
    ExactClassicalTInference
    requires
    relevant sampling assumptions

StudentTConsistency :=
    DegreesOfFreedom
    ∧ PDF
    ∧ CDF
    ∧ Moments
    ∧ SamplingConstruction
    jointly_compatible

InvalidStudentTDistribution :=
    ν≤0
    ∨ PDFNormalisationFailure
    ∨ InvalidMomentClaim
    ∨ IncompatibleConstruction

StudentTValidation :=
    ResolveDegreesOfFreedom
    → ValidateDegreesOfFreedom
    → ResolveSupport
    → ConstructPDF
    → ValidateNormalisation
    → ValidateMomentExistence
    → ResolveSamplingConstruction
    → ValidateInferentialAssumptionsIfApplicable

StudentTDistributionState := {
    RandomVariable,
    DegreesOfFreedom,
    Location,
    Scale,
    Support,
    PDF,
    CDF,
    Moments,
    SamplingStatistic,
    InferenceContext,
    Evidence,
    Confidence,
    Provenance
}

StudentTEvidence :=
    NormalSampleWithUnknownVariance
    ∨ RegressionModel
    ∨ CorrelationModel
    ∨ BayesianPredictiveModel
    ∨ HeavyTailedDataModel
    ∨ TheoreticalDerivation

StudentTUpdate :=
    NewDataDegreesOfFreedomOrModelEvidence
    → UpdateState
    → RecomputePDF
    → RecomputeCDF
    → RecomputeCriticalValues
    → RevalidateMomentsAndInference

SEMANTIC_EQUIVALENCE(StudentTA,StudentTB) :=
    same(DegreesOfFreedom)
    ∧ same(LocationScaleParameters)
    ⇒
    same(StudentTProbabilityMeasure)

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

CoreClasses := {
    StudentsTDistribution,
    ContinuousDistribution,
    TRandomVariable,
    DegreesOfFreedom,
    StudentTPDF,
    TCDF,
    StandardNormal,
    ChiSquaredDistribution,
    SampleMean,
    SampleVariance,
    TStatistic,
    TTest,
    TConfidenceInterval,
    FDistribution,
    StudentTLikelihood,
    StudentTConstraint
}

CoreRelations := {
    distributionOf,
    hasDegreesOfFreedom,
    hasSupport,
    hasPDF,
    hasCDF,
    constructedFrom,
    derivedFrom,
    usedFor,
    convergesTo,
    squaredAs,
    sampledFrom,
    estimatedFrom,
    constrainedBy,
    equivalentTo,
    tracesTo
}

TypedRelations :=
    StudentsTDistribution --distributionOf--> TRandomVariable
    StudentsTDistribution --hasDegreesOfFreedom--> DegreesOfFreedom
    StudentsTDistribution --hasSupport--> Support
    StudentsTDistribution --hasPDF--> StudentTPDF
    StudentsTDistribution --hasCDF--> TCDF
    StudentsTDistribution --constructedFrom--> StandardNormal
    StudentsTDistribution --constructedFrom--> ChiSquaredDistribution
    TStatistic --derivedFrom--> SampleMean
    TStatistic --derivedFrom--> SampleVariance
    StudentsTDistribution --usedFor--> MeanInference
    StudentsTDistribution --convergesTo--> StandardNormal
    TRandomVariable --squaredAs--> FDistribution
    StudentsTDistribution --tracesTo--> Provenance

InferenceRules := {
    Z~N(0,1)
        ∧ V~χ²_ν
        ∧ Z⟂V
        ⇒
        Z/sqrt(V/ν)~t_ν,

    Xi iid N(μ,σ²)
        ⇒
        (X_bar-μ)/(S/√n)
        ~
        t_(n-1),

    T~t_ν
        ∧ ν>1
        ⇒
        E[T]=0,

    T~t_ν
        ∧ ν>2
        ⇒
        Var(T)=ν/(ν-2),

    T~t_ν
        ⇒
        T²~F_(1,ν),

    ν→∞
        ⇒
        t_ν→N(0,1),

    ν≤0
        ⇒
        InvalidStudentTDistribution
}

CanonicalStudentTRepresentation :=
    ParseStudentTModel
    → ResolveRandomVariableOrStatistic
    → ResolveDegreesOfFreedom
    → ResolveLocationScale
    → ValidateParameters
    → ConstructPDF
    → ConstructCDF
    → ResolveMomentExistence
    → ResolveNormalChiSquaredConstruction
    → ResolveInferenceContext
    → ValidateStudentTDistribution

StudentTDistributionGraph :=
    G=(V,E)

V := {
    StudentTDistributions,
    TRandomVariables,
    DegreesOfFreedomNodes,
    PDFs,
    CDFs,
    StandardNormals,
    ChiSquaredDistributions,
    SampleMeans,
    SampleVariances,
    TStatistics,
    TTests,
    ConfidenceIntervals,
    FDistributions,
    BayesianModels,
    Evidence,
    ProvenanceNodes
}

E := {
    distributionOf,
    hasDegreesOfFreedom,
    hasSupport,
    hasPDF,
    hasCDF,
    constructedFrom,
    derivedFrom,
    usedFor,
    convergesTo,
    squaredAs,
    sampledFrom,
    estimatedFrom,
    constrainedBy,
    equivalentTo,
    tracesTo
}

CONFIDENCE(StudentTDistributionResolution) :=
    weighted_mean(
        RandomVariableOrStatisticResolution,
        DegreesOfFreedomResolution,
        LocationScaleResolution,
        PDFConsistency,
        MomentExistenceValidity,
        SamplingConstructionValidity,
        InferentialAssumptionQuality,
        ProvenanceQuality
    )

AI_QUERY_INTERPRETATION :=
    Query
    → DetectStudentTDistributionIntent
    → IdentifyRandomVariableOrTStatistic
    → ResolveDegreesOfFreedom
    → ResolveLocationScaleOrSampleInformation
    → IdentifyRequestedProbabilityCriticalValueOrInference

AI_REASONING_FLOW :=
    Query
    → ParseStudentTProblem
    → ResolveProbabilitySpace
    → ResolveTRandomVariableOrStatistic
    → ResolveDegreesOfFreedom
    → ValidateDegreesOfFreedom
    → ConstructStudentTPDF
    → ConstructOrEvaluateTCDF
    → ComputeTailProbabilityOrCriticalValueIfRequired
    → ValidateMomentExistenceIfRequired
    → ResolveNormalChiSquaredConstruction
    → ResolveOneSampleTwoSamplePairedOrRegressionContextIfRelevant
    → ResolveFDistributionRelationIfRelevant
    → ValidateStudentTModelAndInferenceAssumptions
    → UpdateConfidence
    → PreserveProvenance
    → GenerateExplanation

AI_CONSUMPTION := {
    ParseStudentTDistributionEntities,
    ParseTypedRelations,
    ResolveTRandomVariables,
    ResolveDegreesOfFreedom,
    EvaluateStudentTPDFs,
    EvaluateStudentTCDFs,
    ComputeTailProbabilities,
    ComputeCriticalValues,
    ComputeConfidenceIntervals,
    EvaluateTStatistics,
    ResolveOneSampleTTests,
    ResolveTwoSampleTTests,
    ResolveWelchTTests,
    ResolvePairedTTests,
    ResolveRegressionTStatistics,
    ResolveCorrelationTStatistics,
    ResolveNormalChiSquaredConstruction,
    ResolveFDistributionRelations,
    ResolveBayesianStudentTModels,
    ValidateStudentTDistributions,
    CompareSemanticEquivalence,
    UpdateConfidence,
    PreserveProvenance,
    GenerateFormalExplanation
}

UPDATE_MODEL :=
    NewDataDegreesOfFreedomStatisticOrConstraint
    → ValidateInput
    → IdentifyAffectedStudentTClaims
    → UpdateStudentTState
    → RecomputePDF
    → RecomputeCDF
    → RecomputeCriticalValues
    → ReevaluateMomentExistence
    → RevalidateInferenceRelations
    → RecomputeConfidence
    → PreserveProvenance
    → VersionModel

DegreesOfFreedomUpdate :=
    ChangedDegreesOfFreedom
    ⇒ revalidate(ν>0)
    → recompute(
        PDF,
        CDF,
        VarianceWhereDefined,
        TailProbabilities,
        CriticalValues
    )

SampleUpdate :=
    NewOrChangedSampleData
    ⇒ recompute(
        SampleMean,
        SampleVariance,
        TStatistic,
        DegreesOfFreedom,
        ConfidenceIntervalOrPValue
    )

ModelUpdate :=
    ChangedSamplingOrRegressionAssumptions
    ⇒ reevaluate(
        ExactTDistributionValidity,
        InferentialConfidence
    )

FLOW_HASH_INPUT :=
    canonical(
        ONTOLOGY,
        CoreClasses,
        CoreRelations,
        InferenceRules,
        StudentsTDistribution,
        TConstruction,
        OneSampleTDistribution,
        StudentTValidation,
        SEMANTIC_EQUIVALENCE,
        AI_REASONING_FLOW
    )

FLOW_HASH :=
    HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {
    StudentTDistributionEntityResolution,
    ProbabilitySpace,
    TRandomVariableOrStatistic,
    DegreesOfFreedom,
    Location,
    Scale,
    Support,
    PDF,
    CDF,
    TailProbability,
    CriticalValue,
    MomentExistence,
    SamplingConstruction,
    InferenceContext,
    ConfidenceIntervalOrTestRelation,
    ValidationStatus,
    Confidence,
    Provenance,
    Explanation
}

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA