Probability Distribution Ontology

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

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

BEGIN_FLOW

ProbabilityDistribution :=
    ProbabilityMeasure
    induced_by
    RandomVariable
    ∧ DefinedOn(StateSpace)
    ∧ Normalised
    ∧ Nonnegative

ProbabilitySpace :=
    (Ω,F,P)

RandomVariable :=
    X:Ω→S
    ∧ Measurable(X)

StateSpace :=
    S

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

DistributionMeasure :=
    P_X(B)
    =
    P(X∈B)

LawOfRandomVariable :=
    Law(X)
    =
    P_X

Support :=
    SmallestRelevantSet
    carrying
    distribution probability
    under selected support definition

DiscreteDistribution :=
    ProbabilityDistribution
    with finite_or_countable support

ContinuousDistribution :=
    ProbabilityDistribution
    absolutely_continuous
    with respect to reference measure
    where density exists

MixedDistribution :=
    ProbabilityDistribution
    with discrete_and_continuous components

DegenerateDistribution :=
    ∃c :
        P(X=c)=1

UnivariateDistribution :=
    Law(X)

MultivariateDistribution :=
    Law(X1,...,Xn)

ProbabilityMassFunction :=
    p_X(x)
    =
    P(X=x)

PMF_VALID :=
    ∀x:p_X(x)≥0
    ∧ Σ_x p_X(x)=1

ProbabilityDensityFunction :=
    f_X(x)

PDF_VALID :=
    f_X(x)≥0
    ∧ Integral f_X(x)dx=1

DensityMeasureRelation :=
    P_X(A)
    =
    Integral_A f_X(x)dx

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

CDF_VALID := {
    NonDecreasing,
    RightContinuous,
    LimitAtMinusInfinity=0,
    LimitAtPlusInfinity=1
}

SurvivalFunction :=
    S_X(x)
    =
    P(X>x)
    =
    1-F_X(x)

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

Median :=
    Q_X(0.5)
    under selected convention

Mode :=
    argmax
    PMFOrDensity
    under relevant definition

DistributionParameter :=
    Parameter
    determining
    family member

ParameterVector :=
    θ=(θ1,...,θk)

ParameterSpace :=
    Θ

ParametricDistribution :=
    {P_θ : θ∈Θ}

NonparametricDistribution :=
    Distribution
    not restricted_to
    finite-dimensional parametric family

SemiparametricDistribution :=
    Model
    combining
    finite-dimensional and infinite-dimensional components

LocationParameter :=
    μ

ScaleParameter :=
    σ
    or equivalent

ShapeParameter :=
    α_or_other_shape_value

RateParameter :=
    λ

DegreesOfFreedom :=
    ν

Expectation :=
    E[X]
    when integrable

Variance :=
    Var(X)
    when finite

Moment :=
    E[X^k]
    where exists

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

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

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

ProbabilityGeneratingFunction :=
    G_X(s)
    =
    E[s^X]
    for suitable integer-valued X

LaplaceTransform :=
    L_X(s)
    =
    E[e^(-sX)]
    where defined

DistributionIdentity :=
    Distribution
    uniquely_characterised_by
    CDF

EqualInDistribution :=
    X =_d Y
    iff
    F_X=F_Y

EqualInDistributionMeasure :=
    Law(X)=Law(Y)

AlmostSureEqualityImpliesEqualDistribution :=
    X=Y almost_surely
    ⇒
    X=_dY

EqualDistributionDoesNotImplyAlmostSureEquality :=
    X=_dY
    ↛ X=Y almost_surely

JointDistribution :=
    Law(X,Y)

JointPMF :=
    p_(X,Y)(x,y)

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

JointCDF :=
    F_(X,Y)(x,y)

MarginalDistribution :=
    ProjectionOf(JointDistribution)

MarginalPMF :=
    p_X(x)
    =
    Σ_y p_(X,Y)(x,y)

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

ConditionalDistribution :=
    Law(X|Y)

ConditionalPMF :=
    p_(X|Y)(x|y)
    =
    p_(X,Y)(x,y)/p_Y(y)
    where p_Y(y)>0

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

Independence :=
    X⟂Y
    iff
    Law(X,Y)
    =
    Law(X)⊗Law(Y)

DiscreteIndependence :=
    p_(X,Y)(x,y)
    =
    p_X(x)p_Y(y)

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

Transformation :=
    Y=g(X)

PushforwardDistribution :=
    Law(Y)
    =
    Law(X)∘g^{-1}

DiscreteTransformation :=
    p_Y(y)
    =
    Σ_{x:g(x)=y}
    p_X(x)

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

MultivariateTransformation :=
    Y=g(X)
    → JacobianRule
    under suitable invertibility conditions

Convolution :=
    DistributionOfSum
    of independent random variables

DiscreteConvolution :=
    p_(X+Y)(z)
    =
    Σ_x p_X(x)p_Y(z-x)

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

MixtureDistribution :=
    P
    =
    Σ_i π_iP_i

MixtureWeights :=
    π_i≥0
    ∧ Σ_iπ_i=1

MixturePMF :=
    p(x)
    =
    Σ_i π_i p_i(x)

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

CompoundDistribution :=
    Distribution
    formed_by
    randomising parameter_or_count

HierarchicalDistribution :=
    ConditionalDistribution
    integrated_over
    higher-level distribution

TruncatedDistribution :=
    BaseDistribution
    conditioned_on
    restricted region

CensoredObservationDistribution :=
    DistributionOfObservedCensoredVariable
    induced_by
    censoring mechanism

ShiftedDistribution :=
    Y=X+c

ScaledDistribution :=
    Y=aX

LocationScaleTransformation :=
    Y=μ+σZ

Standardisation :=
    Z
    =
    (X-μ)/σ

DistributionFamily := {
    Bernoulli,
    Binomial,
    Geometric,
    NegativeBinomial,
    Hypergeometric,
    Poisson,
    DiscreteUniform,
    Categorical,
    Multinomial,
    ContinuousUniform,
    Normal,
    Exponential,
    Gamma,
    Beta,
    ChiSquared,
    StudentT,
    FDistribution,
    LogNormal,
    Weibull,
    Cauchy,
    Pareto
}

DistributionClosure :=
    Family
    remains_within
    same family
    under specified operation

NormalAffineClosure :=
    X~Normal
    ⇒ aX+b~Normal
    for a≠0

NormalSumClosure :=
    IndependentNormal(X,Y)
    ⇒ X+Y~Normal

PoissonSumClosure :=
    IndependentPoisson(X,Y)
    ⇒
    X+Y~Poisson(λ_X+λ_Y)

DistributionLimit :=
    SequenceOfDistributions
    → LimitingDistribution

ConvergenceInDistribution :=
    X_n→_dX
    iff
    F_(X_n)(x)→F_X(x)
    at continuity points of F_X

WeakConvergence :=
    ProbabilityMeasures
    converge
    against bounded continuous test functions
    under standard definition

CentralLimitRelation :=
    StandardisedSum
    → NormalDistribution
    under theorem conditions

EmpiricalDistribution :=
    P_n
    =
    (1/n)Σ_i δ_(X_i)

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

SamplingDistribution :=
    Law(T(X1,...,Xn))

Statistic :=
    T(X1,...,Xn)

Likelihood :=
    L(θ|Data)
    =
    Π_i f(x_i|θ)
    for iid density/mass model

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

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

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

DistributionFit :=
    Data
    → CandidateDistribution
    → FitAssessment

GoodnessOfFit :=
    StatisticalAssessment
    of compatibility
    between observed data and distribution

DistributionSelection :=
    CandidateFamilies
    → EstimateParameters
    → EvaluateFit
    → CompareModels

DistributionConstraint :=
    ConstraintOn(
        Support,
        Parameters,
        ProbabilityFunction,
        Moments,
        Dependence
    )

SupportConstraint :=
    ProbabilityOutsideSupport=0

ParameterConstraint :=
    θ∈Θ

NormalisationConstraint :=
    P_X(S)=1

NonnegativityConstraint :=
    P_X(A)≥0
    for measurable A

DistributionConsistency :=
    ProbabilityMeasure
    ∧ Support
    ∧ Parameters
    ∧ PMFOrPDF
    ∧ CDF
    ∧ Moments
    jointly_compatible

DistributionConflict :=
    ViolationOf(
        ProbabilityAxioms
        ∨ SupportConstraint
        ∨ ParameterConstraint
        ∨ NormalisationConstraint
        ∨ DeclaredMomentConstraint
    )

DistributionValidation :=
    ResolveStateSpace
    → ResolveDistributionType
    → ResolveSupport
    → ResolveParameters
    → ValidateProbabilityFunction
    → ValidateCDF
    → ValidateMoments
    → ValidateDependenceStructure

DistributionEvidence :=
    SampleData
    ∨ ModelDefinition
    ∨ TheoreticalDerivation
    ∨ PriorKnowledge
    ∨ ExternalMeasurement

DistributionState := {
    RandomVariable,
    StateSpace,
    DistributionType,
    DistributionFamily,
    Parameters,
    Support,
    PMFOrPDF,
    CDF,
    Moments,
    JointStructure,
    Evidence,
    Confidence,
    Provenance
}

DistributionUpdate :=
    NewDataParameterOrModelEvidence
    → UpdateDistributionState
    → RecomputeProbabilityFunction
    → RecomputeCDF
    → RecomputeMoments
    → Revalidate

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

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

CoreClasses := {
    ProbabilityDistribution,
    ProbabilitySpace,
    RandomVariable,
    StateSpace,
    DiscreteDistribution,
    ContinuousDistribution,
    MixedDistribution,
    PMF,
    PDF,
    CDF,
    Support,
    DistributionParameter,
    DistributionFamily,
    Expectation,
    Variance,
    JointDistribution,
    MarginalDistribution,
    ConditionalDistribution,
    MixtureDistribution
}

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

TypedRelations :=
    ProbabilityDistribution --distributionOf--> RandomVariable
    RandomVariable --distributedAs--> ProbabilityDistribution
    ProbabilityDistribution --hasSupport--> Support
    ProbabilityDistribution --hasParameter--> DistributionParameter
    DiscreteDistribution --hasPMF--> PMF
    ContinuousDistribution --hasPDF--> PDF
    ProbabilityDistribution --hasCDF--> CDF
    ProbabilityDistribution --hasExpectation--> ExpectedValue
    ProbabilityDistribution --hasVariance--> Variance
    MarginalDistribution --marginalOf--> JointDistribution
    ConditionalDistribution --conditionedOn--> RandomVariable
    MixtureDistribution --mixtureOf--> ProbabilityDistribution
    ProbabilityDistribution --tracesTo--> Provenance

InferenceRules := {
    RandomVariable(X)
        ⇒ DistributionExists(X),

    PMF(p)
        ⇒ p(x)≥0
        ∧ Σ_xp(x)=1,

    PDF(f)
        ⇒ f(x)≥0
        ∧ Integral f(x)dx=1,

    Distribution(X)
        ⇒ F_X(x)=P(X≤x),

    X=_dY
        ⇒ Law(X)=Law(Y),

    Independent(X,Y)
        ⇒ Law(X,Y)=Law(X)⊗Law(Y),

    Independent(X,Y)
        ⇒ Law(X+Y)=Convolution(Law(X),Law(Y)),

    ValidDistribution(D)
        ⇒ TotalProbability(D)=1
}

CanonicalProbabilityDistributionRepresentation :=
    ParseDistributionDefinition
    → ResolveRandomVariable
    → ResolveStateSpace
    → ClassifyDistributionType
    → ResolveDistributionFamily
    → ResolveParameters
    → ResolveSupport
    → ResolvePMFOrPDF
    → ResolveCDF
    → ResolveMoments
    → ValidateDistribution

ProbabilityDistributionGraph :=
    G=(V,E)

V := {
    ProbabilitySpaces,
    RandomVariables,
    StateSpaces,
    Distributions,
    DistributionFamilies,
    Parameters,
    Supports,
    PMFs,
    PDFs,
    CDFs,
    Moments,
    JointDistributions,
    Marginals,
    Conditionals,
    Mixtures,
    Evidence,
    ProvenanceNodes
}

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

CONFIDENCE(ProbabilityDistributionResolution) :=
    weighted_mean(
        RandomVariableResolution,
        StateSpaceResolution,
        DistributionTypeResolution,
        FamilyResolution,
        ParameterQuality,
        SupportConsistency,
        ProbabilityFunctionValidity,
        ProvenanceQuality
    )

AI_QUERY_INTERPRETATION :=
    Query
    → DetectProbabilityDistributionConcept
    → IdentifyRandomVariable
    → ResolveStateSpace
    → ResolveKnownProbabilityFunctionOrParameters
    → IdentifyDistributionFamily
    → IdentifyRequestedDistributionProperty

AI_REASONING_FLOW :=
    Query
    → ParseDistributionQuery
    → ResolveProbabilitySpace
    → ResolveRandomVariable
    → ResolveStateSpace
    → DetermineDiscreteContinuousMixedOrDegenerate
    → ResolveSupport
    → ResolveDistributionFamily
    → ResolveParameters
    → ConstructOrValidatePMFOrPDF
    → ConstructOrValidateCDF
    → ComputeMomentsIfRequired
    → ResolveJointMarginalOrConditionalRelationsIfRequired
    → ApplyTransformationConvolutionOrMixtureIfRequired
    → ValidateDistributionConsistency
    → UpdateConfidence
    → PreserveProvenance
    → GenerateExplanation

AI_CONSUMPTION := {
    ParseProbabilityDistributionEntities,
    ParseTypedRelations,
    ResolveRandomVariables,
    ResolveStateSpaces,
    ClassifyDistributionTypes,
    ResolveDistributionFamilies,
    ResolveParameters,
    ResolveSupports,
    EvaluatePMFs,
    EvaluatePDFs,
    EvaluateCDFs,
    ComputeMoments,
    ResolveJointDistributions,
    ComputeMarginals,
    ResolveConditionalDistributions,
    ApplyTransformations,
    ApplyConvolutions,
    ResolveMixtures,
    ValidateDistributions,
    CompareSemanticEquivalence,
    UpdateConfidence,
    PreserveProvenance,
    GenerateFormalExplanation
}

UPDATE_MODEL :=
    NewDataParameterDistributionOrConstraint
    → ValidateInput
    → IdentifyAffectedDistributionClaims
    → UpdateDistributionState
    → RecomputePMFOrPDF
    → RecomputeCDF
    → RecomputeMoments
    → ReevaluateJointMarginalConditionalRelations
    → RevalidateConstraints
    → RecomputeConfidence
    → PreserveProvenance
    → VersionModel

ParameterUpdate :=
    ChangedParameter
    ⇒ recompute(
        ProbabilityFunction,
        CDF,
        Moments,
        Quantiles
    )

DataUpdate :=
    NewSampleData
    ⇒ reevaluate(
        ParameterEstimates,
        DistributionFit,
        CandidateFamilies
    )

SupportUpdate :=
    ChangedSupport
    ⇒ revalidate(
        ProbabilityFunction,
        Normalisation,
        Moments
    )

FLOW_HASH_INPUT :=
    canonical(
        ONTOLOGY,
        CoreClasses,
        CoreRelations,
        InferenceRules,
        Distribution,
        DistributionValidation,
        EqualInDistribution,
        SEMANTIC_EQUIVALENCE,
        AI_REASONING_FLOW
    )

FLOW_HASH :=
    HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {
    ProbabilityDistributionEntityResolution,
    ProbabilitySpace,
    RandomVariable,
    StateSpace,
    DistributionType,
    DistributionFamily,
    Parameters,
    Support,
    PMFOrPDF,
    CDF,
    Moments,
    JointMarginalConditionalStructure,
    ValidationStatus,
    Confidence,
    Provenance,
    Explanation
}

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA