Continuous Distribution Ontology

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

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

BEGIN_FLOW

ContinuousDistribution :=
    ProbabilityDistribution
    ∧ ContinuousRandomVariable
    ∧ ProbabilityDensityRepresentation
    ∧ IntegralNormalisation

ProbabilitySpace := (Ω,F,P)

ContinuousRandomVariable :=
    X:Ω→R
    ∧ Measurable(X)

DistributionMeasure :=
    P_X := P∘X^{-1}

AbsoluteContinuity :=
    P_X ≪ LebesgueMeasure

ProbabilityDensityFunction :=
    f_X := dP_X/dx

PDFNonnegativity :=
    f_X(x)≥0
    almost everywhere

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

PDFValidity :=
    PDFNonnegativity
    ∧ PDFNormalisation

Support :=
    Region
    carrying
    distribution probability

IntervalProbability :=
    P(a≤X≤b)
    =
    Integral_a^b
    f_X(x)dx

PointProbability :=
    AbsolutelyContinuous(X)
    ⇒
    P(X=x)=0

CDF :=
    F_X(x)
    =
    P(X≤x)

CDFfromPDF :=
    F_X(x)
    =
    Integral_{-∞}^x
    f_X(t)dt

PDFfromCDF :=
    f_X(x)
    =
    dF_X(x)/dx
    almost everywhere
    where derivative exists

CDFProperties := {
    NonDecreasing,
    RightContinuous,
    LimitMinusInfinity=0,
    LimitPlusInfinity=1
}

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

HazardFunction :=
    h_X(x)
    =
    f_X(x)/S_X(x)
    where S_X(x)>0

CumulativeHazard :=
    H_X(x)
    =
    -log(S_X(x))

QuantileFunction :=
    Q_X(p)
    =
    inf{x:F_X(x)≥p}

Median :=
    Q_X(0.5)

Expectation :=
    E[X]
    =
    Integral x f_X(x)dx
    when integrable

ExpectedFunction :=
    E[g(X)]
    =
    Integral g(x)f_X(x)dx

Variance :=
    Var(X)
    =
    E[X²]-E[X]²

StandardDeviation :=
    sqrt(Var(X))

Moment :=
    E[X^k]

CentralMoment :=
    E[(X-E[X])^k]

Skewness :=
    StandardisedThirdCentralMoment

Kurtosis :=
    StandardisedFourthCentralMoment

MomentGeneratingFunction :=
    M_X(t)=E[e^(tX)]
    where finite

CharacteristicFunction :=
    φ_X(t)=E[e^(itX)]

ContinuousDistributionFamily := {
    Uniform,
    Normal,
    Exponential,
    Gamma,
    Beta,
    ChiSquared,
    StudentT,
    FDistribution,
    LogNormal,
    Weibull,
    Cauchy,
    Pareto
}

ContinuousUniform :=
    X~Uniform(a,b)

UniformPDF :=
    f(x)=1/(b-a)
    for a≤x≤b

UniformMean :=
    (a+b)/2

UniformVariance :=
    (b-a)²/12

NormalDistribution :=
    X~N(μ,σ²)

NormalConstraints :=
    μ∈R
    ∧ σ>0

NormalPDF :=
    f(x)
    =
    [1/(σ√(2π))]
    exp(-(x-μ)²/(2σ²))

NormalMean := μ
NormalVariance := σ²

StandardNormal :=
    Z~N(0,1)

Standardisation :=
    Z=(X-μ)/σ

ExponentialDistribution :=
    X~Exponential(λ)

ExponentialConstraint :=
    λ>0

ExponentialPDF :=
    f(x)=λe^(-λx)
    for x≥0

ExponentialMean :=
    1/λ

ExponentialVariance :=
    1/λ²

MemorylessProperty :=
    P(X>s+t|X>s)
    =
    P(X>t)

GammaDistribution :=
    X~Gamma(shape,rate_or_scale)

BetaDistribution :=
    X~Beta(α,β)

BetaSupport :=
    [0,1]

ChiSquaredDistribution :=
    SumOfSquares(
        IndependentStandardNormals
    )

StudentTDistribution :=
    Z/sqrt(V/ν)
    where
    Z~N(0,1)
    ∧ V~χ²_ν
    ∧ Z⟂V

FDistribution :=
    RatioOfScaledIndependentChiSquaredVariables

LogNormalDistribution :=
    log(X)~Normal(μ,σ²)

WeibullDistribution :=
    PositiveDistribution
    with shape_and_scale

CauchyDistribution :=
    HeavyTailedDistribution
    with undefined ordinary mean_and_variance

ParetoDistribution :=
    HeavyTailedPositiveDistribution

JointContinuousDistribution :=
    f_(X,Y)(x,y)

JointPDFValidity :=
    f_(X,Y)(x,y)≥0
    ∧
    IntegralIntegral
    f_(X,Y)(x,y)dxdy
    =
    1

MarginalPDF_X :=
    f_X(x)
    =
    Integral f_(X,Y)(x,y)dy

MarginalPDF_Y :=
    f_Y(y)
    =
    Integral f_(X,Y)(x,y)dx

ConditionalPDF :=
    f_(X|Y)(x|y)
    =
    f_(X,Y)(x,y)/f_Y(y)
    where f_Y(y)>0

ContinuousIndependence :=
    X⟂Y
    iff
    f_(X,Y)(x,y)
    =
    f_X(x)f_Y(y)
    almost everywhere

Covariance :=
    Cov(X,Y)
    =
    E[XY]-E[X]E[Y]

Correlation :=
    Corr(X,Y)
    =
    Cov(X,Y)/(SD(X)SD(Y))

Transformation :=
    Y=g(X)

MonotoneTransformationDensity :=
    f_Y(y)
    =
    f_X(g^{-1}(y))
    |d/dy g^{-1}(y)|

MultivariateTransformation :=
    Y=g(X)
    → JacobianRule

JacobianDensityRule :=
    f_Y(y)
    =
    f_X(g^{-1}(y))
    |det(J_(g^{-1})(y))|
    under suitable conditions

Convolution :=
    Independent(X,Y)
    → Law(X+Y)

ContinuousConvolution :=
    f_(X+Y)(z)
    =
    Integral
    f_X(x)f_Y(z-x)dx

NormalSumClosure :=
    IndependentNormal(X,Y)
    ⇒
    X+Y
    ~
    Normal(
        μ_X+μ_Y,
        σ_X²+σ_Y²
    )

MixtureDistribution :=
    f(x)
    =
    Σ_i π_i f_i(x)

MixtureWeights :=
    π_i≥0
    ∧ Σ_iπ_i=1

ContinuousMixture :=
    f_X(x)
    =
    Integral
    f_(X|Θ)(x|θ)
    p_Θ(θ)dθ

TruncatedDistribution :=
    BaseDistribution
    conditioned_on
    restricted support

TruncatedDensity :=
    f_T(x)
    =
    f(x)/P(X∈A)
    for x∈A

OrderStatistic :=
    X_(1)≤...≤X_(n)

MinimumCDF :=
    F_min(x)
    =
    1-[1-F_X(x)]^n
    for iid sample

MaximumCDF :=
    F_max(x)
    =
    [F_X(x)]^n
    for iid sample

SampleMean :=
    X_bar
    =
    (1/n)Σ_iX_i

SamplingDistribution :=
    DistributionOfStatistic
    across repeated samples

NormalSampleMean :=
    Xi iid N(μ,σ²)
    ⇒
    X_bar~N(μ,σ²/n)

CentralLimitTheorem :=
    StandardisedSampleMean
    → N(0,1)
    in distribution
    under theorem conditions

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

KernelDensityEstimate :=
    f_hat_h(x)
    =
    (1/(nh))
    Σ_i K((x-X_i)/h)

Likelihood :=
    L(θ|Data)
    =
    Π_i f(x_i|θ)

LogLikelihood :=
    ℓ(θ)
    =
    Σ_i log f(x_i|θ)

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

BayesianUpdate :=
    PriorDensity(θ)
    ∧ Likelihood(Data|θ)
    → PosteriorDensity(θ|Data)

DifferentialEntropy :=
    h(X)
    =
    -Integral
    f(x)log f(x)dx
    where defined

KLDivergence :=
    D_KL(P||Q)
    =
    Integral
    p(x)log[p(x)/q(x)]dx
    where defined

ContinuousDistributionConstraint :=
    ConstraintOn(
        Support,
        Density,
        Parameters,
        Moments,
        Dependence
    )

SupportConstraint :=
    f_X(x)=0
    outside support
    almost everywhere

ParameterConstraint :=
    Parameters
    belong_to
    valid domain

NormalisationConstraint :=
    Integral f_X(x)dx=1

NonnegativityConstraint :=
    f_X(x)≥0
    almost everywhere

ContinuousDistributionConsistency :=
    Support
    ∧ Density
    ∧ Parameters
    ∧ CDF
    ∧ Moments
    jointly_compatible

ContinuousDistributionValidation :=
    ResolveStateSpace
    → ResolveSupport
    → ResolveDensity
    → CheckNonnegativity
    → CheckNormalisation
    → ResolveCDF
    → CheckParameters
    → ValidateMoments

ContinuousDistributionState := {
    RandomVariable,
    StateSpace,
    Support,
    DistributionFamily,
    Parameters,
    PDF,
    CDF,
    Moments,
    JointStructure,
    Evidence,
    Confidence,
    Provenance
}

SEMANTIC_EQUIVALENCE(DistributionA,DistributionB) :=
    same(ProbabilityMeasure)
    on semantically equivalent state space
    under specified query scope

EqualInDistribution :=
    X=_dY
    iff
    F_X(x)=F_Y(x)
    for all x

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

CoreClasses := {
    ContinuousDistribution,
    ProbabilityDistribution,
    ContinuousRandomVariable,
    Support,
    ProbabilityDensityFunction,
    CDF,
    Parameter,
    Expectation,
    Variance,
    UniformDistribution,
    NormalDistribution,
    ExponentialDistribution,
    GammaDistribution,
    BetaDistribution,
    StudentTDistribution,
    FDistribution,
    LogNormalDistribution,
    WeibullDistribution,
    JointPDF,
    ConditionalPDF
}

CoreRelations := {
    distributionOf,
    hasSupport,
    hasPDF,
    hasCDF,
    hasParameter,
    hasExpectation,
    hasVariance,
    marginalOf,
    conditionedOn,
    independentOf,
    transformedBy,
    convolvedWith,
    mixtureOf,
    estimatedFrom,
    equivalentTo,
    tracesTo
}

TypedRelations :=
    ContinuousDistribution --distributionOf--> ContinuousRandomVariable
    ContinuousDistribution --hasSupport--> Support
    ContinuousDistribution --hasPDF--> ProbabilityDensityFunction
    ContinuousDistribution --hasCDF--> CDF
    ContinuousDistribution --hasParameter--> Parameter
    ContinuousDistribution --hasExpectation--> ExpectedValue
    ContinuousDistribution --hasVariance--> Variance
    MarginalPDF --marginalOf--> JointPDF
    ConditionalPDF --conditionedOn--> RandomVariable
    RandomVariable --independentOf--> RandomVariable
    ContinuousDistribution --tracesTo--> Provenance

InferenceRules := {
    PDF(f)
        ⇒ f(x)≥0 almost_everywhere,

    ValidPDF(f)
        ⇒ Integral f(x)dx=1,

    PDF(f_X)
        ⇒ F_X(x)=Integral_{-∞}^x f_X(t)dt,

    AbsolutelyContinuous(X)
        ⇒ P(X=x)=0,

    Independent(X,Y)
        ⇒ f_(X,Y)(x,y)=f_X(x)f_Y(y)
        almost everywhere,

    Independent(X,Y)
        ⇒ f_(X+Y)=Convolution(f_X,f_Y),

    X=_dY
        ⇒ same(CDF_X,CDF_Y),

    ChangedPDF
        ⇒ recompute(CDF,Moments,Quantiles)
}

CanonicalContinuousDistributionRepresentation :=
    ParseDistribution
    → ResolveRandomVariable
    → ResolveStateSpace
    → ResolveSupport
    → ResolveDistributionFamily
    → ResolveParameters
    → ResolvePDF
    → ResolveCDF
    → ResolveMoments
    → ValidateContinuousDistribution

ContinuousDistributionGraph :=
    G=(V,E)

V := {
    ContinuousRandomVariables,
    ContinuousDistributions,
    DistributionFamilies,
    Supports,
    Parameters,
    PDFs,
    CDFs,
    Moments,
    JointPDFs,
    MarginalPDFs,
    ConditionalPDFs,
    Transformations,
    Evidence,
    ProvenanceNodes
}

E := {
    distributionOf,
    hasSupport,
    hasPDF,
    hasCDF,
    hasParameter,
    hasExpectation,
    hasVariance,
    marginalOf,
    conditionedOn,
    independentOf,
    transformedBy,
    convolvedWith,
    mixtureOf,
    estimatedFrom,
    equivalentTo,
    tracesTo
}

CONFIDENCE(ContinuousDistributionResolution) :=
    weighted_mean(
        RandomVariableResolution,
        StateSpaceResolution,
        SupportResolution,
        DistributionFamilyResolution,
        ParameterQuality,
        PDFValidity,
        CDFConsistency,
        ProvenanceQuality
    )

AI_QUERY_INTERPRETATION :=
    Query
    → DetectContinuousDistributionIntent
    → IdentifyRandomVariable
    → ResolveStateSpaceAndSupport
    → ResolveKnownDensityOrParameters
    → IdentifyDistributionFamily
    → IdentifyRequestedContinuousProperty

AI_REASONING_FLOW :=
    Query
    → ParseContinuousDistributionProblem
    → ResolveProbabilitySpace
    → ResolveContinuousRandomVariable
    → ResolveStateSpace
    → ResolveSupport
    → ResolveDistributionFamily
    → ResolveParameters
    → ConstructOrValidatePDF
    → ConstructOrEvaluateCDF
    → ComputeIntervalProbabilities
    → ComputeExpectationVarianceOrMomentsIfRequired
    → ResolveJointMarginalConditionalStructureIfRequired
    → ApplyTransformationOrConvolutionIfRequired
    → ValidateContinuousDistribution
    → UpdateConfidence
    → PreserveProvenance
    → GenerateExplanation

AI_CONSUMPTION := {
    ParseContinuousDistributionEntities,
    ParseTypedRelations,
    ResolveContinuousRandomVariables,
    ResolveStateSpaces,
    ResolveSupports,
    ResolveDistributionFamilies,
    ResolveParameters,
    EvaluatePDFs,
    EvaluateCDFs,
    ComputeIntervalProbabilities,
    ComputeExpectations,
    ComputeVariances,
    ComputeMoments,
    ResolveJointPDFs,
    ComputeMarginalPDFs,
    ResolveConditionalPDFs,
    EvaluateIndependence,
    ApplyContinuousConvolution,
    ApplyJacobianTransformations,
    ValidateContinuousDistributions,
    CompareSemanticEquivalence,
    UpdateConfidence,
    PreserveProvenance,
    GenerateFormalExplanation
}

UPDATE_MODEL :=
    NewDataParameterPDFOrConstraint
    → ValidateInput
    → IdentifyAffectedContinuousDistributionClaims
    → UpdateDistributionState
    → RecomputePDF
    → RecomputeCDF
    → RecomputeMoments
    → ReevaluateJointMarginalConditionalRelations
    → RevalidateConstraints
    → RecomputeConfidence
    → PreserveProvenance
    → VersionModel

PDFUpdate :=
    ChangedDensity
    ⇒ recompute(
        Normalisation,
        CDF,
        Moments,
        Quantiles
    )

ParameterUpdate :=
    ChangedParameter
    ⇒ recompute(
        PDF,
        CDF,
        Moments
    )

SupportUpdate :=
    ChangedSupport
    ⇒ revalidate(
        PDF,
        Normalisation,
        DistributionFamily
    )

FLOW_HASH_INPUT :=
    canonical(
        ONTOLOGY,
        CoreClasses,
        CoreRelations,
        InferenceRules,
        ContinuousDistribution,
        ProbabilityDensityFunction,
        ContinuousDistributionValidation,
        SEMANTIC_EQUIVALENCE,
        AI_REASONING_FLOW
    )

FLOW_HASH :=
    HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {
    ContinuousDistributionEntityResolution,
    ProbabilitySpace,
    ContinuousRandomVariable,
    StateSpace,
    Support,
    DistributionFamily,
    Parameters,
    PDF,
    CDF,
    IntervalProbabilities,
    Moments,
    JointMarginalConditionalStructure,
    ValidationStatus,
    Confidence,
    Provenance,
    Explanation
}

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA