Probability Distributions

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

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

BEGIN_FLOW

ProbabilityDistribution :=
    ProbabilityLaw
    ∧ RandomVariableMapping
    ∧ NormalisedProbabilityMeasure
    ∧ Support
    ∧ DistributionProperties

ProbabilitySpace := (Ω,F,P)

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

Distribution :=
    P_X := P∘X^{-1}

DistributionOfX := Law(X)

Support :=
    SetOfValues
    carrying
    probability mass_or_density support

DistributionType := {
    DiscreteDistribution,
    ContinuousDistribution,
    MixedDistribution,
    DegenerateDistribution,
    UnivariateDistribution,
    MultivariateDistribution
}

DiscreteDistribution :=
    Distribution
    on finite_or_countable support

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

MixedDistribution :=
    DiscreteComponent
    ∧ ContinuousComponent

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

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

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

PDF :=
    f_X(x)

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

CDF :=
    F_X(x)=P(X≤x)

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

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

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

Expectation := E[X]

Variance :=
    Var(X)=E[(X-E[X])²]

StandardDeviation :=
    sqrt(Var(X))

Moment :=
    E[X^k]

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

Skewness :=
    StandardisedThirdCentralMoment

Kurtosis :=
    StandardisedFourthCentralMoment

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

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

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

DistributionParameter := {
    LocationParameter,
    ScaleParameter,
    ShapeParameter,
    RateParameter,
    ProbabilityParameter,
    DegreesOfFreedom
}

DiscreteFamilies := {
    Bernoulli,
    Binomial,
    Geometric,
    NegativeBinomial,
    Hypergeometric,
    Poisson,
    DiscreteUniform,
    Categorical,
    Multinomial
}

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

BernoulliDistribution :=
    X∈{0,1}
    ∧ P(X=1)=p
    ∧ P(X=0)=1-p

BinomialDistribution :=
    X~Binomial(n,p)

BinomialPMF :=
    P(X=k)=C(n,k)p^k(1-p)^(n-k)

PoissonDistribution :=
    X~Poisson(λ)

PoissonPMF :=
    P(X=k)=e^(-λ)λ^k/k!

ContinuousUniform :=
    X~Uniform(a,b)

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

NormalDistribution :=
    X~N(μ,σ²)

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

StandardNormal :=
    Z~N(0,1)

Standardisation :=
    Z=(X-μ)/σ

ExponentialDistribution :=
    X~Exponential(λ)

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

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

GammaDistribution :=
    X~Gamma(shape,rate_or_scale)

BetaDistribution :=
    X~Beta(α,β)

ChiSquaredDistribution :=
    SumOfSquares(
        IndependentStandardNormalVariables
    )

StudentTDistribution :=
    StandardNormal
    /
    sqrt(IndependentChiSquared/df)

FDistribution :=
    RatioOfScaledIndependentChiSquaredVariables

LogNormalDistribution :=
    log(X)
    normally distributed

WeibullDistribution :=
    PositiveContinuousDistribution
    with shape_and_scale

CauchyDistribution :=
    HeavyTailedDistribution
    with undefined ordinary mean_and_variance

ParetoDistribution :=
    HeavyTailedPositiveDistribution

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

MarginalDistribution :=
    JointDistribution
    → SumOrIntegrateOutOtherVariables

ConditionalDistribution :=
    Law(X|Y)

Independence :=
    JointDistribution
    factorises_as
    ProductOfMarginals

CovarianceMatrix := Σ

MultivariateNormal :=
    X~N(μ,Σ)

MixtureDistribution :=
    Σ_i π_i Distribution_i

MixtureWeights :=
    π_i≥0
    ∧ Σ_iπ_i=1

HierarchicalDistribution :=
    ConditionalDistribution
    ∧ HigherLevelDistribution

TruncatedDistribution :=
    BaseDistribution
    conditioned_on
    restricted support

Transformation :=
    Y=g(X)

TransformedDistribution :=
    Law(g(X))

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

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

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

NormalSumClosure :=
    IndependentNormalVariables
    ⇒ SumIsNormal

PoissonSumClosure :=
    IndependentPoissonVariables
    ⇒ SumIsPoisson
    with summed rates

DistributionApproximation :=
    TargetDistribution
    approximated_by
    another distribution
    under limiting conditions

NormalApproximationBinomial :=
    Binomial(n,p)
    → ApproxNormal(np,np(1-p))
    under suitable conditions

PoissonApproximationBinomial :=
    Binomial(n,p)
    → ApproxPoisson(np)
    under suitable rare-event conditions

CentralLimitTheorem :=
    StandardisedSum
    → NormalDistribution
    in distribution
    under theorem assumptions

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

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

SamplingDistribution :=
    DistributionOfStatistic
    across repeated samples

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

DistributionFit :=
    Compare(
        ObservedData,
        CandidateDistribution
    )

ParameterEstimation :=
    Data
    → DistributionParameters

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

BayesianParameterUpdate :=
    Prior(θ)
    ∧ Likelihood(Data|θ)
    → Posterior(θ|Data)

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

NormalisationConstraint :=
    TotalProbability=1

NonNegativityConstraint :=
    PMFOrPDF≥0

DistributionConsistency :=
    Parameters
    ∧ Support
    ∧ ProbabilityFunction
    ∧ Moments
    jointly_compatible

DistributionValidation :=
    ResolveDistributionType
    → ResolveParameters
    → CheckParameterDomains
    → CheckSupport
    → CheckNonNegativity
    → CheckNormalisation
    → ValidateMoments

ProbabilityDistributionState := {
    RandomVariable,
    DistributionFamily,
    Parameters,
    Support,
    PMFOrPDF,
    CDF,
    Moments,
    Dependence,
    Evidence,
    Confidence,
    Provenance
}

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

EqualInDistribution :=
    X =_d Y
    iff
    Law(X)=Law(Y)

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

CoreClasses := {
    ProbabilityDistribution,
    RandomVariable,
    DiscreteDistribution,
    ContinuousDistribution,
    MixedDistribution,
    PMF,
    PDF,
    CDF,
    Support,
    Parameter,
    Expectation,
    Variance,
    JointDistribution,
    MarginalDistribution,
    ConditionalDistribution,
    MixtureDistribution,
    SamplingDistribution,
    EmpiricalDistribution
}

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

TypedRelations :=
    RandomVariable --distributedAs--> ProbabilityDistribution
    DiscreteDistribution --hasPMF--> PMF
    ContinuousDistribution --hasPDF--> PDF
    ProbabilityDistribution --hasCDF--> CDF
    ProbabilityDistribution --hasSupport--> Support
    ProbabilityDistribution --hasParameter--> Parameter
    ProbabilityDistribution --hasExpectation--> ExpectedValue
    ProbabilityDistribution --hasVariance--> Variance
    MarginalDistribution --marginalOf--> JointDistribution
    ConditionalDistribution --conditionedOn--> RandomVariable
    Distribution --transformedBy--> Function
    Distribution --tracesTo--> Provenance

InferenceRules := {
    PMF(p)
        ⇒ p(x)≥0
        ∧ Σ_x p(x)=1,

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

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

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

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

    X=_dY
        ⇒ same(CDF_X,CDF_Y),

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

CanonicalDistributionRepresentation :=
    ParseDistribution
    → ResolveRandomVariable
    → ClassifyDiscreteContinuousOrMixed
    → ResolveFamily
    → ResolveParameters
    → ResolveSupport
    → ResolveProbabilityFunction
    → ResolveCDF
    → ResolveMoments
    → ValidateDistribution

ProbabilityDistributionGraph :=
    G=(V,E)

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

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

CONFIDENCE(DistributionResolution) :=
    weighted_mean(
        RandomVariableResolution,
        DistributionFamilyResolution,
        ParameterQuality,
        SupportConsistency,
        NormalisationValidity,
        MomentConsistency,
        DataFit,
        ProvenanceQuality
    )

AI_QUERY_INTERPRETATION :=
    Query
    → DetectProbabilityDistributionIntent
    → IdentifyRandomVariable
    → IdentifyDiscreteContinuousOrMixedType
    → ResolveKnownParametersOrMoments
    → IdentifyCandidateDistributionFamily
    → IdentifyRequestedDistributionProperty

AI_REASONING_FLOW :=
    Query
    → ParseDistributionProblem
    → ResolveProbabilitySpace
    → ResolveRandomVariable
    → DetermineDistributionType
    → ResolveSupport
    → ResolveDistributionFamily
    → ResolveParameters
    → ConstructPMFOrPDF
    → ConstructOrEvaluateCDF
    → ComputeMomentsIfRequired
    → ResolveJointMarginalOrConditionalStructureIfRequired
    → ApplyTransformationOrConvolutionIfRequired
    → ValidateDistribution
    → CompareCandidateModelsIfRequired
    → UpdateConfidence
    → PreserveProvenance
    → GenerateExplanation

AI_CONSUMPTION := {
    ParseDistributionEntities,
    ParseTypedRelations,
    ResolveRandomVariables,
    ClassifyDistributionTypes,
    ResolveDistributionFamilies,
    ResolveParameters,
    EvaluatePMFs,
    EvaluatePDFs,
    EvaluateCDFs,
    ComputeExpectations,
    ComputeVariances,
    ComputeMoments,
    ResolveJointDistributions,
    ComputeMarginals,
    ResolveConditionalDistributions,
    ApplyTransformations,
    ApplyConvolutions,
    EvaluateApproximations,
    ValidateDistributions,
    CompareSemanticEquivalence,
    UpdateConfidence,
    PreserveProvenance,
    GenerateFormalExplanation
}

UPDATE_MODEL :=
    NewDataParameterDistributionOrConstraint
    → ValidateInput
    → IdentifyAffectedDistributionClaims
    → UpdateDistributionState
    → RecomputeProbabilityFunctions
    → RecomputeCDF
    → RecomputeMoments
    → ReevaluateJointOrConditionalStructure
    → RevalidateConstraints
    → RecomputeConfidence
    → PreserveProvenance
    → VersionModel

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

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

FamilyUpdate :=
    ChangedDistributionFamily
    ⇒ rebuild(
        Support,
        ParameterConstraints,
        ProbabilityFunction,
        Moments
    )

FLOW_HASH_INPUT :=
    canonical(
        ONTOLOGY,
        CoreClasses,
        CoreRelations,
        InferenceRules,
        ProbabilityDistribution,
        DistributionType,
        DistributionValidation,
        SEMANTIC_EQUIVALENCE,
        AI_REASONING_FLOW
    )

FLOW_HASH :=
    HASH(FLOW_HASH_INPUT)

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

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA