Cauchy Distribution Ontology

FULL EXECUTABLE FLOW ALGEBRA — CAUCHY DISTRIBUTION ONTOLOGY
Kireth Research — Semantic Library
URL: https://kirethresearch.org/booksai-core/ontologies/probability-statistics-ontologies/probability-distributions/cauchy-distribution-ontology.html

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

BEGIN_FLOW

CauchyDistribution :=
    ContinuousProbabilityDistribution
    ∧ RealLineSupport
    ∧ LocationParameter
    ∧ PositiveScaleParameter
    ∧ SymmetricHeavyTailedDistribution
    ∧ StableDistribution
    ∧ UndefinedOrdinaryMeanAndVariance

ProbabilitySpace := (Ω,F,P)

CauchyRandomVariable :=
    X~Cauchy(x0,γ)

LocationParameter :=
    x0

ScaleParameter :=
    γ

ParameterConstraints :=
    x0∈R
    ∧ γ>0

Support :=
    R

StandardCauchy :=
    Z~Cauchy(0,1)

CauchyPDF :=
    f_X(x)
    =
    1
    /
    [
        πγ
        (
            1+
            ((x-x0)/γ)^2
        )
    ]

EquivalentPDF :=
    f_X(x)
    =
    γ
    /
    [
        π(
            (x-x0)^2+γ²
        )
    ]

PDFNonnegativity :=
    f_X(x)>0
    for all x∈R

PDFNormalisation :=
    Integral_{-∞}^{∞}
    f_X(x)dx
    =
    1

CauchyCDF :=
    F_X(x)
    =
    1/π
    arctan(
        (x-x0)/γ
    )
    +
    1/2

SurvivalFunction :=
    S_X(x)
    =
    1-F_X(x)

QuantileFunction :=
    Q(p)
    =
    x0
    +
    γ tan[
        π(p-1/2)
    ]
    for 0<p<1

Median :=
    x0

Mode :=
    x0

SymmetryCentre :=
    x0

Symmetry :=
    f_X(x0-d)
    =
    f_X(x0+d)

InterquartileRange :=
    Q(0.75)-Q(0.25)

CauchyIQR :=
    2γ

ScaleFromIQR :=
    γ
    =
    IQR/2

HalfWidthAtHalfMaximum :=
    γ

FullWidthAtHalfMaximum :=
    2γ

Mean :=
    Undefined

OrdinaryExpectation :=
    DoesNotExist

Variance :=
    Undefined

StandardDeviation :=
    Undefined

HigherMoments :=
    OrdinaryMoments
    generally undefined
    for positive integer orders

AbsoluteMomentExistence :=
    E[|X|^p]
    finite
    only for suitable p<1
    in standard centred case

PrincipalValueMean :=
    CauchyPrincipalValue
    may equal x0
    under symmetric interpretation
    but is not ordinary expectation

HeavyTail :=
    f_X(x)
    asymptotically
    proportional_to
    1/x²

TailProbabilityDecay :=
    P(|X|>x)
    asymptotically
    proportional_to
    1/x

CharacteristicFunction :=
    φ_X(t)
    =
    exp(
        ix0t
        -
        γ|t|
    )

MomentGeneratingFunction :=
    DoesNotExist
    for nonzero real t

StableDistributionRelation :=
    CauchyDistribution
    is
    α-stable
    with stability index 1

CauchyStability :=
    IndependentCauchyVariables
    under linear combinations
    remain Cauchy
    with transformed location_and_scale

IndependentCauchySum :=
    X~Cauchy(x1,γ1)
    ∧ Y~Cauchy(x2,γ2)
    ∧ X⟂Y
    ⇒
    X+Y
    ~
    Cauchy(
        x1+x2,
        γ1+γ2
    )

AffineTransformation :=
    X~Cauchy(x0,γ)
    ⇒
    aX+b
    ~
    Cauchy(
        ax0+b,
        |a|γ
    )
    for a≠0

Standardisation :=
    Z
    =
    (X-x0)/γ

StandardisationResult :=
    Z~Cauchy(0,1)

NormalRatioConstruction :=
    U~N(0,1)
    ∧ V~N(0,1)
    ∧ U⟂V
    ⇒
    U/V
    ~
    Cauchy(0,1)

ScaledNormalRatio :=
    x0
    +
    γ(U/V)
    ~
    Cauchy(x0,γ)

StudentTRelation :=
    t_1
    ≡
    StandardCauchy

StudentTSpecialCase :=
    StudentTDistribution(df=1)
    ⇒
    CauchyDistribution(0,1)

InverseTransformSampling :=
    U~Uniform(0,1)
    ⇒
    X
    =
    x0
    +
    γ tan[
        π(U-1/2)
    ]
    ~
    Cauchy(x0,γ)

TangentUniformRelation :=
    Θ~Uniform(-π/2,π/2)
    ⇒
    tanΘ
    ~
    StandardCauchy

LorentzianRelation :=
    CauchyPDF
    also known_as
    LorentzianLineShape
    in physical contexts

BreitWignerRelation :=
    CertainBreitWignerForms
    correspond_to
    CauchyOrLorentzianStructure
    under parameter mapping

ResonanceLineShape :=
    CauchyTypeDensity
    may model
    resonance peaks
    under appropriate physical assumptions

SampleMeanBehaviour :=
    iid Cauchy samples
    ⇒
    SampleMean
    has same Cauchy distribution
    as individual observations
    for standard equal-scale case

NoLawOfLargeNumbersForOrdinaryMean :=
    OrdinarySampleMean
    does_not_converge
    to finite population expectation
    because expectation does not exist

SampleMedian :=
    RobustLocationEstimator
    for Cauchy location

MedianConsistency :=
    SampleMedian
    may consistently estimate
    location parameter
    under regular conditions

CauchyLikelihood :=
    X_1,...,X_n
    iid Cauchy(x0,γ)

Likelihood :=
    L(x0,γ|Data)
    =
    Π_i
    [
        γ
        /
        π(
            (x_i-x0)^2+γ²
        )
    ]

LogLikelihood :=
    ℓ(x0,γ)
    =
    nlogγ
    -
    nlogπ
    -
    Σ_i
    log(
        (x_i-x0)^2+γ²
    )

MaximumLikelihoodEstimation :=
    GenerallyRequires
    numerical optimisation
    for unknown location and scale

RobustStatisticsRelation :=
    CauchyDistribution
    illustrates
    failure of moment-based summaries
    and importance of robust location_scale measures

OutlierBehaviour :=
    HeavyTails
    assign
    substantially more probability
    to extreme values
    than GaussianDistribution

CauchyVsNormal :=
    Both symmetric
    but Cauchy
    has much heavier tails
    and undefined mean_and_variance

Entropy :=
    h(X)
    =
    ln(
        4πγ
    )

KLDivergenceRelation :=
    Closed-form expressions
    may exist
    between Cauchy distributions
    under specified parameters

CauchyProcessRelation :=
    StableLevyProcess
    with Cauchy increments
    under standard construction

CauchyNoise :=
    HeavyTailedNoiseModel
    with Cauchy-distributed errors

BayesianCauchyPrior :=
    CauchyDistribution
    may be used
    as heavy-tailed prior
    for parameters

HalfCauchyDistribution :=
    |X|
    where
    X~Cauchy(0,γ)

HalfCauchySupport :=
    [0,∞)

HalfCauchyPriorRelation :=
    HalfCauchy
    commonly used
    as heavy-tailed prior
    for positive scale parameters
    in some Bayesian models

CauchyDistributionConstraint :=
    ConstraintOn(
        Location,
        Scale,
        Support,
        PDF,
        UndefinedMoments,
        StableStructure
    )

LocationConstraint :=
    x0∈R

ScaleConstraint :=
    γ>0

SupportConstraint :=
    X∈R

NormalisationConstraint :=
    Integral_{-∞}^{∞}
    f_X(x)dx
    =
    1

MomentConstraint :=
    DoNotAssign
    finite ordinary mean_or_variance

CauchyConsistency :=
    Location
    ∧ Scale
    ∧ PDF
    ∧ CDF
    ∧ Quantiles
    ∧ UndefinedMomentStructure
    jointly_compatible

InvalidCauchyDistribution :=
    γ≤0
    ∨ PDFNormalisationFailure
    ∨ CDFMismatch
    ∨ FiniteMeanClaim
    ∨ FiniteVarianceClaim

CauchyValidation :=
    ResolveLocation
    → ResolveScale
    → ValidateScale
    → ResolveSupport
    → ConstructPDF
    → ValidateNormalisation
    → ConstructCDF
    → ValidateQuantiles
    → RejectInvalidMomentClaims

CauchyDistributionState := {
    RandomVariable,
    Location,
    Scale,
    Support,
    PDF,
    CDF,
    QuantileFunction,
    Median,
    Mode,
    MomentExistence,
    StableRelations,
    Evidence,
    Confidence,
    Provenance
}

CauchyEvidence :=
    HeavyTailedData
    ∨ RatioOfNormalVariables
    ∨ ResonanceModel
    ∨ StableProcessModel
    ∨ BayesianPriorSpecification
    ∨ TheoreticalDerivation

CauchyUpdate :=
    NewDataLocationScaleOrModelEvidence
    → UpdateCauchyState
    → RecomputePDF
    → RecomputeCDF
    → RecomputeQuantiles
    → RevalidateMomentClaims
    → RevalidateStableRelations

SEMANTIC_EQUIVALENCE(CauchyA,CauchyB) :=
    same(Location)
    ∧ same(Scale)
    ⇒
    same(CauchyProbabilityMeasure)

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

CoreClasses := {
    CauchyDistribution,
    ContinuousDistribution,
    CauchyRandomVariable,
    LocationParameter,
    ScaleParameter,
    CauchyPDF,
    CauchyCDF,
    QuantileFunction,
    Median,
    Mode,
    StableDistribution,
    StudentTDistribution,
    NormalDistribution,
    LorentzianDistribution,
    HalfCauchyDistribution,
    CauchyConstraint
}

CoreRelations := {
    distributionOf,
    hasLocation,
    hasScale,
    hasSupport,
    hasPDF,
    hasCDF,
    hasQuantile,
    equivalentTo,
    specialCaseOf,
    constructedFrom,
    stableUnder,
    relatedTo,
    estimatedFrom,
    constrainedBy,
    tracesTo
}

TypedRelations :=
    CauchyDistribution --distributionOf--> CauchyRandomVariable
    CauchyDistribution --hasLocation--> LocationParameter
    CauchyDistribution --hasScale--> ScaleParameter
    CauchyDistribution --hasSupport--> Support
    CauchyDistribution --hasPDF--> CauchyPDF
    CauchyDistribution --hasCDF--> CauchyCDF
    CauchyDistribution --hasQuantile--> QuantileFunction
    StandardCauchy --equivalentTo--> StudentTDistribution(df=1)
    CauchyDistribution --constructedFrom--> RatioOfIndependentNormals
    CauchyDistribution --stableUnder--> AffineCombination
    CauchyDistribution --relatedTo--> LorentzianDistribution
    CauchyDistribution --tracesTo--> Provenance

InferenceRules := {
    X~Cauchy(x0,γ)
        ⇒
        F_X(x)
        =
        1/π arctan((x-x0)/γ)+1/2,

    X~Cauchy(x0,γ)
        ⇒
        Median(X)=x0,

    X~Cauchy(x0,γ)
        ⇒
        Mode(X)=x0,

    X~Cauchy(x0,γ)
        ⇒
        OrdinaryMean(X)=Undefined,

    X~Cauchy(x0,γ)
        ⇒
        Variance(X)=Undefined,

    U,V iid N(0,1)
        ⇒
        U/V~Cauchy(0,1),

    T~t_1
        ⇒
        T~Cauchy(0,1),

    Independent(
        X~Cauchy(x1,γ1),
        Y~Cauchy(x2,γ2)
    )
        ⇒
        X+Y~Cauchy(x1+x2,γ1+γ2),

    γ≤0
        ⇒
        InvalidCauchyDistribution
}

CanonicalCauchyRepresentation :=
    ParseCauchyModel
    → ResolveRandomVariable
    → ResolveLocation
    → ResolveScale
    → ValidateScale
    → ResolveSupport
    → ConstructPDF
    → ConstructCDF
    → ConstructQuantileFunction
    → ResolveMomentExistence
    → ResolveStableNormalRatioAndStudentTRelations
    → ValidateCauchyDistribution

CauchyDistributionGraph :=
    G=(V,E)

V := {
    CauchyDistributions,
    CauchyRandomVariables,
    LocationParameters,
    ScaleParameters,
    PDFs,
    CDFs,
    Quantiles,
    Medians,
    Modes,
    StableDistributions,
    StudentTDistributions,
    NormalRatioConstructions,
    LorentzianRelations,
    HalfCauchyDistributions,
    Evidence,
    ProvenanceNodes
}

E := {
    distributionOf,
    hasLocation,
    hasScale,
    hasSupport,
    hasPDF,
    hasCDF,
    hasQuantile,
    equivalentTo,
    specialCaseOf,
    constructedFrom,
    stableUnder,
    relatedTo,
    estimatedFrom,
    constrainedBy,
    tracesTo
}

CONFIDENCE(CauchyDistributionResolution) :=
    weighted_mean(
        RandomVariableResolution,
        LocationResolution,
        ScaleResolution,
        SupportValidity,
        PDFConsistency,
        CDFConsistency,
        MomentExistenceValidity,
        ProvenanceQuality
    )

AI_QUERY_INTERPRETATION :=
    Query
    → DetectCauchyDistributionIntent
    → IdentifyHeavyTailedRandomVariable
    → ResolveLocation
    → ResolveScale
    → IdentifyRequestedProbabilityQuantileOrProperty
    → ResolveNormalRatioStudentTOrStableContext

AI_REASONING_FLOW :=
    Query
    → ParseCauchyDistributionProblem
    → ResolveProbabilitySpace
    → ResolveCauchyRandomVariable
    → ResolveLocation
    → ResolveScale
    → ValidateScale
    → ConstructCauchyPDF
    → ConstructCauchyCDF
    → ComputeRequestedIntervalOrTailProbability
    → ComputeQuantileMedianModeOrIQRIfRequired
    → ValidateUndefinedMeanAndVariance
    → ResolveNormalRatioConstructionIfRelevant
    → ResolveStudentTOrStableDistributionRelationIfRelevant
    → ValidateCauchyModel
    → UpdateConfidence
    → PreserveProvenance
    → GenerateExplanation

AI_CONSUMPTION := {
    ParseCauchyDistributionEntities,
    ParseTypedRelations,
    ResolveCauchyRandomVariables,
    ResolveLocationParameters,
    ResolveScaleParameters,
    EvaluateCauchyPDFs,
    EvaluateCauchyCDFs,
    ComputeIntervalProbabilities,
    ComputeTailProbabilities,
    ComputeQuantiles,
    ComputeMedians,
    ComputeModes,
    ComputeIQRs,
    ValidateMomentNonexistence,
    ApplyInverseTransformSampling,
    ResolveNormalRatioConstruction,
    ResolveStudentTRelations,
    ResolveStableDistributionRelations,
    ResolveLorentzianRelations,
    EstimateCauchyParameters,
    ValidateCauchyDistributions,
    CompareSemanticEquivalence,
    UpdateConfidence,
    PreserveProvenance,
    GenerateFormalExplanation
}

UPDATE_MODEL :=
    NewDataLocationScaleOrConstraint
    → ValidateInput
    → IdentifyAffectedCauchyClaims
    → UpdateCauchyState
    → RecomputePDF
    → RecomputeCDF
    → RecomputeQuantiles
    → RevalidateMomentExistence
    → ReevaluateStableRelations
    → RecomputeConfidence
    → PreserveProvenance
    → VersionModel

LocationUpdate :=
    ChangedLocation
    ⇒ recompute(
        PDFLocation,
        CDF,
        Quantiles,
        Median,
        Mode
    )

ScaleUpdate :=
    ChangedScale
    ⇒ revalidate(γ>0)
    → recompute(
        PDFScale,
        CDF,
        Quantiles,
        IQR
    )

DataUpdate :=
    NewHeavyTailedData
    ⇒ reevaluate(
        LocationEstimate,
        ScaleEstimate,
        ModelFit,
        Confidence
    )

FLOW_HASH_INPUT :=
    canonical(
        ONTOLOGY,
        CoreClasses,
        CoreRelations,
        InferenceRules,
        CauchyDistribution,
        CauchyPDF,
        NormalRatioConstruction,
        CauchyValidation,
        SEMANTIC_EQUIVALENCE,
        AI_REASONING_FLOW
    )

FLOW_HASH :=
    HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {
    CauchyDistributionEntityResolution,
    ProbabilitySpace,
    CauchyRandomVariable,
    Location,
    Scale,
    Support,
    PDF,
    CDF,
    QuantileFunction,
    RequestedProbability,
    Median,
    Mode,
    IQR,
    MeanExistenceStatus,
    VarianceExistenceStatus,
    StableDistributionRelations,
    ValidationStatus,
    Confidence,
    Provenance,
    Explanation
}

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA