Gamma Distribution Ontology

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

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

BEGIN_FLOW

GammaDistribution :=
    ContinuousProbabilityDistribution
    ∧ PositiveSupport
    ∧ ShapeParameter
    ∧ RateOrScaleParameter
    ∧ WaitingTimeModel
    ∧ ExponentialFamilyMember

ProbabilitySpace := (Ω,F,P)

GammaRandomVariable :=
    X~Gamma(α,β)
    under shape-rate parameterisation

ShapeParameter := α
RateParameter := β
ScaleParameter := θ

ParameterConstraints :=
    α>0
    ∧ β>0

RateScaleRelation :=
    θ=1/β

Support :=
    (0,∞)

GammaFunction :=
    Γ(α)
    =
    Integral_0^∞
    t^(α-1)e^(-t)dt

GammaRecurrence :=
    Γ(α+1)
    =
    αΓ(α)

IntegerGamma :=
    Γ(n)
    =
    (n-1)!
    for positive integer n

GammaPDFRate :=
    f_X(x)
    =
    [β^α/Γ(α)]
    x^(α-1)
    e^(-βx)
    for x>0

GammaPDFScale :=
    f_X(x)
    =
    [1/(Γ(α)θ^α)]
    x^(α-1)
    e^(-x/θ)
    for x>0

PDFOutsideSupport :=
    f_X(x)=0
    for x≤0

PDFNonnegativity :=
    f_X(x)≥0

PDFNormalisation :=
    Integral_0^∞
    f_X(x)dx
    =
    1

GammaCDF :=
    F_X(x)
    =
    P(X≤x)

LowerIncompleteGamma :=
    γ(α,z)
    =
    Integral_0^z
    t^(α-1)e^(-t)dt

RegularisedLowerGamma :=
    P(α,z)
    =
    γ(α,z)/Γ(α)

GammaCDFRate :=
    F_X(x)
    =
    P(α,βx)

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

UpperIncompleteGamma :=
    Γ(α,z)
    =
    Integral_z^∞
    t^(α-1)e^(-t)dt

RegularisedUpperGamma :=
    Q(α,z)
    =
    Γ(α,z)/Γ(α)

GammaSurvivalRate :=
    S_X(x)
    =
    Q(α,βx)

Expectation :=
    E[X]
    =
    α/β
    =
    αθ

Variance :=
    Var(X)
    =
    α/β²
    =
    αθ²

StandardDeviation :=
    sqrt(α)/β
    =
    sqrt(α)θ

Mode :=
    (α-1)/β
    for α≥1
    with boundary behaviour when α≤1

Skewness :=
    2/sqrt(α)

ExcessKurtosis :=
    6/α

Moment :=
    E[X^k]
    =
    Γ(α+k)
    /
    [Γ(α)β^k]
    where defined

MomentGeneratingFunction :=
    M_X(t)
    =
    (β/(β-t))^α
    for t<β

CharacteristicFunction :=
    φ_X(t)
    =
    (β/(β-it))^α

LaplaceTransform :=
    L_X(s)
    =
    (β/(β+s))^α
    for s>-β

ExponentialRelation :=
    Gamma(shape=1,rate=β)
    ≡
    Exponential(β)

ErlangDistribution :=
    GammaDistribution
    with integer shape
    α=n

ErlangWaitingTime :=
    SumOfNIndependentExponentialWaitingTimes

PoissonProcessRelation :=
    HomogeneousPoissonProcess(rate=β)
    ⇒
    WaitingTimeToNthEvent
    ~
    Gamma(n,β)

WaitingTimeToNthEvent :=
    T_n
    =
    Σ_{i=1..n}E_i

ExponentialComponents :=
    E_i iid Exponential(β)

GammaAdditivity :=
    X~Gamma(α1,β)
    ∧ Y~Gamma(α2,β)
    ∧ X⟂Y
    ⇒
    X+Y
    ~
    Gamma(α1+α2,β)

GammaScaleAdditivity :=
    CommonScale θ
    ∧ IndependentGammaComponents
    ⇒
    ShapesAdd

DifferentRateSums :=
    GammaVariables
    with unequal rates
    generally do_not_form
    simple GammaDistribution

ScalingProperty :=
    X~Gamma(α,β)
    ∧ c>0
    ⇒
    cX
    ~
    Gamma(α,β/c)
    in rate parameterisation

StandardGamma :=
    X~Gamma(α,1)

RateStandardisation :=
    βX
    ~
    Gamma(α,1)

ChiSquaredRelation :=
    χ²_ν
    ≡
    Gamma(
        α=ν/2,
        β=1/2
    )

ChiSquaredScaling :=
    If X~Gamma(α,β)
    then
    2βX
    ~
    χ²_(2α)
    where degrees-of-freedom interpretation is valid

BetaGammaRelation :=
    X~Gamma(α,1)
    ∧ Y~Gamma(β,1)
    ∧ X⟂Y
    ⇒
    X/(X+Y)
    ~
    Beta(α,β)

IndependentSumAndRatio :=
    X/(X+Y)
    independent_of
    X+Y
    under common-rate independent gamma construction

GammaRatioRelation :=
    RatiosOfIndependentGammaVariables
    generate
    BetaAndFDistributionStructures
    under parameter conditions

HazardFunction :=
    h_X(x)
    =
    f_X(x)/S_X(x)

HazardShape :=
    α=1
    ⇒ ConstantHazard

IncreasingHazard :=
    α>1
    generally
    ⇒ increasing hazard
    for GammaDistribution

DecreasingHazard :=
    0<α<1
    generally
    ⇒ decreasing hazard

GammaLifetimeModel :=
    PositiveLifetime
    with flexible hazard behaviour

ReliabilityFunction :=
    R(t)
    =
    S_X(t)

MeanResidualLife :=
    E[X-t|X>t]
    where defined

GammaLikelihood :=
    X_1,...,X_n
    iid Gamma(α,β)

LogLikelihoodRate :=
    ℓ(α,β)
    =
    nαlogβ
    -
    nlogΓ(α)
    +
    (α-1)Σ_i log x_i
    -
    βΣ_i x_i

KnownShapeRateMLE :=
    β_hat
    =
    nα/Σ_i x_i

KnownRateShapeEstimation :=
    Requires
    solving equations involving
    digamma function
    for general MLE

DigammaFunction :=
    ψ(α)
    =
    d/dα logΓ(α)

MethodOfMoments :=
    SampleMean=m
    ∧ SampleVariance=v
    ⇒
    α_hat
    =
    m²/v

MethodOfMomentsRate :=
    β_hat
    =
    m/v

MethodOfMomentsScale :=
    θ_hat
    =
    v/m

GammaPriorPoissonRate :=
    λ~Gamma(α,β)

GammaPoissonConjugacy :=
    GammaPrior
    ∧ PoissonLikelihood
    → GammaPosterior

PoissonPosteriorShape :=
    α_post
    =
    α+Σ_i x_i

PoissonPosteriorRate :=
    β_post
    =
    β+Σ_i t_i
    for exposure t_i

GammaPriorExponentialRate :=
    λ~Gamma(α,β)
    ∧ ExponentialWaitingTimes
    → GammaPosterior

ExponentialPosteriorShape :=
    α_post
    =
    α+n

ExponentialPosteriorRate :=
    β_post
    =
    β+Σ_i x_i

BayesianPredictiveRelation :=
    IntegratingPoissonLikelihood
    over GammaRate
    yields
    NegativeBinomialTypePredictive

GammaMixture :=
    Distribution
    formed_by
    mixing GammaComponents

GammaProcessRelation :=
    GammaProcess
    has
    GammaDistributedIncrements
    under standard definition

GammaProcess :=
    IncreasingLevyProcess
    with independent increments

GammaRegressionRelation :=
    PositiveContinuousResponse
    may be modelled
    using GammaGeneralisedLinearModel

GammaGLM :=
    Response
    GammaDistributed
    with mean linked_to
    linear predictor

GammaEntropy :=
    h(X)
    =
    α
    -
    logβ
    +
    logΓ(α)
    +
    (1-α)ψ(α)
    under shape-rate parameterisation

GammaDistributionConstraint :=
    ConstraintOn(
        Shape,
        RateOrScale,
        Support,
        PDF,
        Moments
    )

ShapeConstraint :=
    α>0

RateConstraint :=
    β>0

ScaleConstraint :=
    θ>0

SupportConstraint :=
    X>0

NormalisationConstraint :=
    Integral_0^∞f_X(x)dx=1

ParameterisationConsistency :=
    θ=1/β

GammaConsistency :=
    Shape
    ∧ RateOrScale
    ∧ PDF
    ∧ CDF
    ∧ Moments
    jointly_compatible

InvalidGammaDistribution :=
    α≤0
    ∨ β≤0
    ∨ θ≤0
    ∨ RateScaleMismatch
    ∨ PDFNormalisationFailure

GammaValidation :=
    ResolveShape
    → ResolveRateOrScale
    → CanonicaliseParameterisation
    → ValidateParameters
    → ResolveSupport
    → ConstructPDF
    → ValidateNormalisation
    → ConstructCDF
    → ValidateMoments

GammaDistributionState := {
    RandomVariable,
    Shape,
    Rate,
    Scale,
    Support,
    PDF,
    CDF,
    SurvivalFunction,
    HazardFunction,
    Mean,
    Variance,
    ProcessRelations,
    Evidence,
    Confidence,
    Provenance
}

GammaEvidence :=
    WaitingTimeData
    ∨ PositiveContinuousData
    ∨ PoissonProcessModel
    ∨ BayesianPriorSpecification
    ∨ TheoreticalDerivation

GammaUpdate :=
    NewDataShapeRateScaleOrModelEvidence
    → UpdateGammaState
    → CanonicaliseParameterisation
    → RecomputePDF
    → RecomputeCDF
    → RecomputeMoments
    → Revalidate

SEMANTIC_EQUIVALENCE(GammaA,GammaB) :=
    same(Shape)
    ∧ same(CanonicalRate)
    ⇒
    same(GammaProbabilityMeasure)

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

CoreClasses := {
    GammaDistribution,
    ContinuousDistribution,
    GammaRandomVariable,
    ShapeParameter,
    RateParameter,
    ScaleParameter,
    GammaFunction,
    GammaPDF,
    GammaCDF,
    SurvivalFunction,
    HazardFunction,
    ErlangDistribution,
    ExponentialDistribution,
    ChiSquaredDistribution,
    PoissonProcess,
    GammaLikelihood,
    GammaConstraint
}

CoreRelations := {
    distributionOf,
    hasShape,
    hasRate,
    hasScale,
    hasSupport,
    hasPDF,
    hasCDF,
    hasMean,
    hasVariance,
    specialCaseOf,
    sumOf,
    waitingTimeFor,
    conjugatePriorFor,
    estimatedFrom,
    constrainedBy,
    equivalentTo,
    tracesTo
}

TypedRelations :=
    GammaDistribution --distributionOf--> GammaRandomVariable
    GammaDistribution --hasShape--> ShapeParameter
    GammaDistribution --hasRate--> RateParameter
    GammaDistribution --hasScale--> ScaleParameter
    GammaDistribution --hasSupport--> Support
    GammaDistribution --hasPDF--> GammaPDF
    GammaDistribution --hasCDF--> GammaCDF
    ExponentialDistribution --specialCaseOf--> GammaDistribution
    ErlangDistribution --specialCaseOf--> GammaDistribution
    ChiSquaredDistribution --specialCaseOf--> GammaDistribution
    GammaRandomVariable --waitingTimeFor--> PoissonProcessEvent
    GammaDistribution --tracesTo--> Provenance

InferenceRules := {
    X~Gamma(α,β)
        ⇒
        E[X]=α/β,

    X~Gamma(α,β)
        ⇒
        Var(X)=α/β²,

    Gamma(1,β)
        ⇒ Exponential(β),

    Independent(
        X~Gamma(α1,β),
        Y~Gamma(α2,β)
    )
        ⇒
        X+Y~Gamma(α1+α2,β),

    PoissonProcess(rate=β)
        ⇒
        WaitingTimeToNthEvent
        ~Gamma(n,β),

    X~Gamma(α,β)
        ⇒
        2βX~χ²_(2α)
        where applicable,

    α≤0
        ∨ β≤0
        ⇒ InvalidGammaDistribution
}

CanonicalGammaRepresentation :=
    ParseGammaModel
    → ResolveRandomVariable
    → ResolveShape
    → ResolveRateOrScale
    → CanonicaliseToShapeRate
    → ResolveSupport
    → ConstructPDF
    → ConstructCDF
    → ComputeMoments
    → ResolveSpecialCaseRelations
    → ValidateGammaDistribution

GammaDistributionGraph :=
    G=(V,E)

V := {
    GammaDistributions,
    GammaRandomVariables,
    ShapeParameters,
    RateParameters,
    ScaleParameters,
    GammaFunctions,
    PDFs,
    CDFs,
    SurvivalFunctions,
    Hazards,
    ErlangDistributions,
    ExponentialDistributions,
    ChiSquaredDistributions,
    PoissonProcesses,
    Likelihoods,
    Evidence,
    ProvenanceNodes
}

E := {
    distributionOf,
    hasShape,
    hasRate,
    hasScale,
    hasSupport,
    hasPDF,
    hasCDF,
    hasMean,
    hasVariance,
    specialCaseOf,
    sumOf,
    waitingTimeFor,
    conjugatePriorFor,
    estimatedFrom,
    constrainedBy,
    equivalentTo,
    tracesTo
}

CONFIDENCE(GammaDistributionResolution) :=
    weighted_mean(
        RandomVariableResolution,
        ShapeResolution,
        RateScaleResolution,
        ParameterisationConsistency,
        PDFConsistency,
        MomentConsistency,
        ProcessRelationValidity,
        ProvenanceQuality
    )

AI_QUERY_INTERPRETATION :=
    Query
    → DetectGammaDistributionIntent
    → IdentifyPositiveRandomVariable
    → ResolveShape
    → ResolveRateOrScale
    → IdentifyRequestedProbabilityOrProperty
    → ResolveWaitingTimeOrBayesianContext

AI_REASONING_FLOW :=
    Query
    → ParseGammaProblem
    → ResolveProbabilitySpace
    → ResolveGammaRandomVariable
    → ResolveShape
    → ResolveRateOrScale
    → CanonicaliseParameterisation
    → ValidateParameters
    → ConstructGammaPDF
    → ConstructOrEvaluateGammaCDF
    → ComputeRequestedProbability
    → ComputeMeanVarianceHazardOrMomentIfRequired
    → ResolveExponentialErlangOrChiSquaredRelationIfRelevant
    → ResolvePoissonProcessOrBayesianRelationIfRelevant
    → ValidateGammaModel
    → UpdateConfidence
    → PreserveProvenance
    → GenerateExplanation

AI_CONSUMPTION := {
    ParseGammaDistributionEntities,
    ParseTypedRelations,
    ResolveGammaRandomVariables,
    ResolveShapeParameters,
    ResolveRateParameters,
    ResolveScaleParameters,
    CanonicaliseGammaParameterisations,
    EvaluateGammaFunctions,
    EvaluateGammaPDFs,
    EvaluateGammaCDFs,
    ComputeProbabilities,
    ComputeMeans,
    ComputeVariances,
    ComputeHazards,
    ResolveExponentialRelations,
    ResolveErlangRelations,
    ResolveChiSquaredRelations,
    ResolvePoissonProcessWaitingTimes,
    EstimateGammaParameters,
    ApplyGammaConjugateUpdates,
    ValidateGammaDistributions,
    CompareSemanticEquivalence,
    UpdateConfidence,
    PreserveProvenance,
    GenerateFormalExplanation
}

UPDATE_MODEL :=
    NewDataShapeRateScaleOrConstraint
    → ValidateInput
    → IdentifyAffectedGammaClaims
    → UpdateGammaState
    → CanonicaliseParameterisation
    → RecomputePDF
    → RecomputeCDF
    → RecomputeSurvivalAndHazard
    → RecomputeMoments
    → RevalidateSpecialCaseRelations
    → RecomputeConfidence
    → PreserveProvenance
    → VersionModel

ShapeUpdate :=
    ChangedShape
    ⇒ revalidate(α>0)
    → recompute(
        PDF,
        CDF,
        Mean,
        Variance,
        HazardShape
    )

RateUpdate :=
    ChangedRate
    ⇒ revalidate(β>0)
    → recompute(
        Scale,
        PDF,
        CDF,
        Mean,
        Variance
    )

ScaleUpdate :=
    ChangedScale
    ⇒ revalidate(θ>0)
    → β=1/θ
    → RateUpdate

DataUpdate :=
    NewPositiveContinuousData
    ⇒ reevaluate(
        ParameterEstimates,
        ModelFit,
        PosteriorIfBayesian
    )

FLOW_HASH_INPUT :=
    canonical(
        ONTOLOGY,
        CoreClasses,
        CoreRelations,
        InferenceRules,
        GammaDistribution,
        GammaPDFRate,
        GammaAdditivity,
        GammaValidation,
        SEMANTIC_EQUIVALENCE,
        AI_REASONING_FLOW
    )

FLOW_HASH :=
    HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {
    GammaDistributionEntityResolution,
    ProbabilitySpace,
    GammaRandomVariable,
    Shape,
    Rate,
    Scale,
    Support,
    PDF,
    CDF,
    SurvivalFunction,
    HazardFunction,
    RequestedProbability,
    Mean,
    Variance,
    SpecialCaseRelations,
    ProcessRelations,
    ValidationStatus,
    Confidence,
    Provenance,
    Explanation
}

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA