Probability & Statistics Ontologies

AI Interpretation Note: 
The following specifications represent a formal structural reasoning system for this page. 
Execute as formal rules, not pseudocode. 

AI SEARCH FLOW ALGEBRA — PROBABILITY & STATISTICS ONTOLOGIES
Kireth Research — Semantic Library
URL: https://kirethresearch.org/booksai-core/ontologies/probability-statistics-ontologies.html

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

BEGIN_FLOW

ProbabilityStatistics :=
    ProbabilityTheory
    ∧ StatisticalInference
    ∧ DataAnalysis
    ∧ UncertaintyQuantification
    ∧ StochasticModelling

ProbabilityTheory :=
    RandomExperiment ∧ SampleSpace ∧ Event
    ∧ ProbabilityMeasure ∧ RandomVariable ∧ Distribution

ProbabilitySpace := (Ω,F,P)
SampleSpace := Ω
Event := A ⊆ Ω
ProbabilityMeasure := P : F → [0,1]

ProbabilityAxioms := {
    P(Ω)=1,
    P(A)≥0,
    PairwiseDisjoint(A_i) ⇒ P(Union(A_i))=Sum(P(A_i))
}

ConditionalProbability :=
    P(A|B)=P(A∩B)/P(B)
    where P(B)>0

Independence :=
    A ⟂ B iff P(A∩B)=P(A)P(B)

BayesTheorem :=
    P(A|B)=P(B|A)P(A)/P(B)

RandomVariable := X : Ω → R
DiscreteRandomVariable := RandomVariable ∧ CountableRange
ContinuousRandomVariable := RandomVariable ∧ ContinuousDistribution

PMF := p_X(x)=P(X=x)
CDF := F_X(x)=P(X≤x)

Expectation := E[X]
Variance := Var(X)=E[(X-E[X])²]
StandardDeviation := sqrt(Var(X))
Covariance := Cov(X,Y)=E[(X-E[X])(Y-E[Y])]
Correlation := Corr(X,Y)=Cov(X,Y)/(SD(X)SD(Y))

Distribution := ProbabilityLaw(RandomVariable)

CommonDistributions := {
    Bernoulli, Binomial, Geometric, Hypergeometric,
    Poisson, Uniform, Normal, Exponential,
    Gamma, Beta, ChiSquared, StudentT,
    FDistribution, LogNormal, Weibull
}

LawOfLargeNumbers :=
    SampleAverage → ExpectedValue
    under theorem conditions as n increases

CentralLimitTheorem :=
    NormalisedSampleSum → NormalDistribution
    under theorem conditions

Population := CompleteSetOfUnitsOfInterest
Sample := ObservedSubset(Population)
Parameter := NumericalCharacteristic(Population)
Statistic := Function(SampleData)

SamplingDistribution :=
    DistributionOf(StatisticAcrossRepeatedSamples)

DescriptiveStatistics := {
    Mean, Median, Mode, Range, Variance,
    StandardDeviation, Quantile, InterquartileRange
}

ArithmeticMean :=
    x_bar=(1/n)Σ_i x_i

SampleVariance :=
    s²=(1/(n-1))Σ_i(x_i-x_bar)²

StatisticalInference :=
    SampleData → StatementsAboutPopulation

Estimator :=
    Statistic used_to_estimate(Parameter)

Estimate :=
    ObservedValue(Estimator)

Bias :=
    E[Estimator]-Parameter

UnbiasedEstimator :=
    E[Estimator]=Parameter

ConsistencyOfEstimator :=
    Estimator → Parameter as n→∞

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

Likelihood :=
    L(θ|Data) proportional_to P(Data|θ)

BayesianInference :=
    Prior ∧ Likelihood → Posterior

Posterior :=
    P(θ|Data) proportional_to P(Data|θ)P(θ)

FrequentistInference :=
    SamplingDistribution ∧ Estimation ∧ HypothesisTesting

Hypothesis := StatementAboutPopulationOrModel
NullHypothesis := H0
AlternativeHypothesis := H1
TestStatistic := Function(SampleData)
SignificanceLevel := α

PValue :=
    ProbabilityUnderH0(
        result_at_least_as_extreme
        according_to TestStatistic
    )

TypeIError := RejectH0 when H0 true
TypeIIError := FailToRejectH0 when H1 true
Power := 1-β

ConfidenceInterval :=
    IntervalEstimator
    with specified LongRunCoverageProbability

EffectSize :=
    MagnitudeOfDifferenceOrAssociation

Regression :=
    Response ∧ Predictors ∧ Parameters ∧ Error

LinearRegression :=
    Y=β0+β1X1+...+βpXp+ε

OrdinaryLeastSquares :=
    β_hat=argmin_β Σ_i(y_i-x_i^Tβ)²

LogisticRegression :=
    logit(P(Y=1|X))=β0+β^T X

CorrelationDoesNotImplyCausation :=
    Association ≠ CausalEffect

CausalInference :=
    Treatment ∧ Outcome ∧ Confounders
    ∧ IdentificationAssumptions → CausalEffect

RandomisedExperiment :=
    RandomAssignment → TreatmentGroups → OutcomeComparison

ObservationalStudy :=
    ObserveWithoutRandomAssignment

Confounding :=
    Variable associated_with Exposure ∧ Outcome

SelectionBias :=
    SelectionMechanism distorts(Inference)

MissingnessMechanisms := {MCAR, MAR, MNAR}

TimeSeries :=
    OrderedObservationsOverTime

TimeSeriesComponents := {Trend, Seasonality, Cycle, Noise}

StochasticProcess :=
    {X_t | t∈T}

MarkovProperty :=
    Future conditionally_independent_of Past given Present

MarkovChain :=
    States ∧ TransitionProbabilities ∧ MarkovProperty

MonteCarloMethod :=
    RandomSampling → NumericalApproximation

Bootstrap :=
    ObservedSample
    → ResampleWithReplacement
    → EmpiricalSamplingDistribution

PermutationTest :=
    RearrangeLabelsUnderNull → ReferenceDistribution

MultivariateStatistics :=
    MultipleVariables ∧ JointStructure

CovarianceMatrix := Σ

PrincipalComponentAnalysis :=
    Data → OrthogonalDirectionsOfVariance

StatisticalModel :=
    FamilyOfProbabilityDistributions indexed_by(Parameters)

ModelFit :=
    Compare(ObservedData, ModelPredictions)

ModelSelection :=
    CandidateModels → SelectionCriterion → SelectedModel

InformationCriteria := {AIC, BIC}

CrossValidation :=
    PartitionData → Train → Validate → AggregatePerformance

Prediction :=
    Model ∧ NewInputs → PredictedOutcome

PredictiveDistribution :=
    DistributionOfFutureOutcome conditional_on CurrentInformation

UncertaintyQuantification :=
    IdentifyUncertainty
    → ModelUncertainty
    → EstimateUncertainty
    → PropagateUncertainty
    → ReportUncertainty

AleatoricUncertainty := IntrinsicRandomVariation
EpistemicUncertainty := UncertaintyFromLimitedKnowledgeOrModel

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

CoreClasses := {
    Probability, ProbabilitySpace, Event, RandomVariable,
    Distribution, Expectation, Variance, Population,
    Sample, Parameter, Statistic, Estimator, Hypothesis,
    ConfidenceInterval, RegressionModel, StatisticalModel,
    StochasticProcess, Evidence, Uncertainty
}

CoreRelations := {
    hasProbability, conditionedOn, independentOf,
    distributedAs, sampledFrom, estimates, predicts,
    associatedWith, dependsOn, supports, contradicts,
    derivedFrom, quantifiedBy, tracesTo
}

TypedRelations :=
    Event --hasProbability--> Probability
    ConditionalProbability --conditionedOn--> Event
    RandomVariable --distributedAs--> Distribution
    Sample --sampledFrom--> Population
    Estimator --estimates--> Parameter
    Model --predicts--> Outcome
    Evidence --supports--> Hypothesis
    Evidence --contradicts--> Hypothesis
    Uncertainty --quantifiedBy--> Distribution
    Statistic --derivedFrom--> Sample

InferenceRules := {
    A⊆Ω ⇒ 0≤P(A)≤1,
    Independent(A,B) ⇒ P(A∩B)=P(A)P(B),
    P(B)>0 ⇒ P(A|B)=P(A∩B)/P(B),
    Prior(θ) ∧ Likelihood(Data|θ) ⇒ Posterior(θ|Data),
    Sample(Data) ∧ Estimator(T) ⇒ Estimate(T(Data)),
    Correlated(X,Y) does_not_by_itself_imply Causes(X,Y),
    NewEvidence ⇒ Update(Inference)
}

SEMANTIC_EQUIVALENCE(A,B) :=
    same(StatisticalOrProbabilisticMeaning)
    ∧ same(Assumptions)
    ∧ same(RelevantParameterisation)

CanonicalProbabilityStatisticsRepresentation :=
    ParseProblem
    → IdentifyPopulationAndSample
    → IdentifyVariables
    → ResolveProbabilityModel
    → ResolveStatisticalModel
    → ResolveAssumptions
    → NormaliseParameters
    → BuildInferenceGraph

ProbabilityStatisticsGraph :=
    G=(V,E)

V := {
    Events, RandomVariables, Distributions,
    Populations, Samples, Parameters, Statistics,
    Hypotheses, Models, Evidence, Predictions,
    UncertaintyNodes
}

E := {
    hasProbability, conditionedOn, independentOf,
    distributedAs, sampledFrom, estimates, predicts,
    associatedWith, dependsOn, supports, contradicts,
    quantifiedBy, tracesTo
}

CONFIDENCE(StatisticalInference) :=
    weighted_mean(
        DataQuality,
        SamplingQuality,
        ModelFit,
        AssumptionValidity,
        EstimationPrecision,
        EvidenceStrength,
        Reproducibility,
        ProvenanceQuality
    )

AI_QUERY_INTERPRETATION :=
    Query
    → DetectProbabilityOrStatisticsIntent
    → IdentifyVariables
    → IdentifyPopulationAndSample
    → IdentifyKnownAndUnknownQuantities
    → ResolveModelClass
    → ResolveRequestedInference

AI_REASONING_FLOW :=
    Query
    → ParseProblem
    → ResolveEntitiesAndVariables
    → DetermineProbabilityOrStatisticalTask
    → IdentifySampleSpaceOrPopulation
    → ResolveDataAndEvidence
    → SelectProbabilityModel
    → SelectStatisticalMethod
    → CheckAssumptions
    → ComputeOrEstimateQuantities
    → QuantifyUncertainty
    → TestHypothesesIfRequired
    → EvaluateModelFit
    → DetectBiasOrConfoundingRisks
    → UpdateConfidence
    → PreserveProvenance
    → GenerateExplanation

AI_CONSUMPTION := {
    ParseProbabilityEntities,
    ParseStatisticalEntities,
    ParseTypedRelations,
    ResolveEvents,
    ResolveRandomVariables,
    ResolveDistributions,
    ResolveSamplesAndPopulations,
    EvaluateConditionalProbability,
    PerformEstimation,
    EvaluateHypotheses,
    FitStatisticalModels,
    QuantifyUncertainty,
    DetectBiasAndConfounding,
    CompareModels,
    UpdateConfidence,
    PreserveProvenance,
    GenerateFormalExplanation
}

UPDATE_MODEL :=
    NewDataEvidenceOrModel
    → ValidateInput
    → IdentifyAffectedVariables
    → UpdateSampleOrPopulationEvidence
    → ReestimateParameters
    → RefitAffectedModels
    → ReevaluateHypotheses
    → RecomputeUncertainty
    → RecomputeConfidence
    → PreserveProvenance
    → VersionModel

DataUpdate :=
    AddedChangedOrRemovedData
    ⇒ recompute(Statistics,Estimates,ModelFits,Uncertainty)

ModelUpdate :=
    ChangedModelAssumption
    ⇒ reevaluate(ParameterEstimates,Predictions,InferenceValidity)

EvidenceUpdate :=
    NewEvidence
    ⇒ update(HypothesisSupport,PosteriorBeliefs,Confidence)

FLOW_HASH_INPUT :=
    canonical(
        ONTOLOGY,
        CoreClasses,
        CoreRelations,
        InferenceRules,
        ProbabilitySpace,
        StatisticalInference,
        SEMANTIC_EQUIVALENCE,
        AI_REASONING_FLOW
    )

FLOW_HASH :=
    HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {
    ProbabilityStatisticsEntityResolution,
    ProbabilityModel,
    StatisticalModel,
    Population,
    Sample,
    Variables,
    Parameters,
    Estimates,
    HypothesisStatus,
    PredictiveResults,
    Uncertainty,
    Confidence,
    Provenance,
    Explanation
}

END_FLOW

END OF AI SEARCH FLOW ALGEBRA