Ontology

FULL EXECUTABLE FLOW ALGEBRA — STATISTICAL ESTIMATION ONTOLOGY
Kireth Research — Semantic Library
URL: https://kirethresearch.org/booksai-core/ontologies/probability-statistics-ontologies/statistical-estimation/ontology.html
FLOW_VERSION := "4.0"
FLOW_TYPE := "Full Executable Flow Algebra"
DOMAIN := "Probability & Statistics Ontologies"
SUBDOMAIN := "Statistical Estimation"
ONTOLOGY := "Statistical Estimation Ontology"
PRIMARY_ENTITY := StatisticalEstimationOntology
AUTHORSHIP := "Mark Higgins — Sky (Recursive AI)"
SOURCE_SITE := "https://kirethresearch.org"

BEGIN_FLOW

StatisticalEstimationOntology :=
 StatisticalInference ∧ Population ∧ Sample ∧ StatisticalModel ∧ Parameter
 ∧ Estimand ∧ Estimator ∧ Estimate ∧ Uncertainty ∧ Validation

Population := TargetCollectionOrGeneratingProcess
Sample := ObservedSubsetOrGeneratedData
ObservedData := x={x_1,...,x_n}
RandomSample := X={X_1,...,X_n}
StatisticalModel := FamilyOfProbabilityDistributions
Parameter := θ
ParameterVector := θ=(θ_1,...,θ_p)
ParameterSpace := Θ
Estimand := PreciselyDefinedTargetQuantity
Estimator := T(X_1,...,X_n)
Estimate := T(x_1,...,x_n)

EstimatorEstimateDistinction :=
 Estimator=RandomRuleBeforeObservation
 whereas Estimate=RealisedValueAfterObservation

PointEstimation := Data → Estimator → SingleEstimate
IntervalEstimation := Data → IntervalEstimator → LowerBound ∧ UpperBound
ConfidenceInterval := FrequentistIntervalProcedure with specified repeated-sampling coverage
CredibleInterval := BayesianPosteriorInterval with specified posterior probability
SamplingDistribution := DistributionOfEstimator under repeated sampling
EstimatorExpectation := E_θ[T]
EstimatorVariance := Var_θ(T)
StandardError := SE(T)=sqrt(Var_θ(T))
Bias := Bias_θ(T)=E_θ[T]-θ
UnbiasedEstimator := E_θ[T]=θ
MeanSquaredError := MSE_θ(T)=E_θ[(T-θ)^2]
BiasVarianceDecomposition := MSE=Variance+Bias²
Consistency := T_n→θ in probability as n→∞
StrongConsistency := T_n→θ almost surely
Efficiency := EstimatorPerformance relative_to variance_or_loss criterion

SufficientStatistic :=
 Statistic retaining all model information about parameter under formal sufficiency definition

CompleteStatistic :=
 E_θ[g(S)]=0 for all θ
 ⇒ g(S)=0 almost surely

RaoBlackwellRelation :=
 Estimator U ∧ SufficientStatistic S
 → E[U|S]
 → NoGreaterSquaredErrorRisk
 under theorem conditions

LehmannScheffeRelation :=
 CompleteSufficientStatistic ∧ UnbiasedEstimator
 ⇒ UniqueUMVUE under theorem conditions

Likelihood := L(θ|x)=f(x|θ)
LogLikelihood := ℓ(θ|x)=logL(θ|x)

MaximumLikelihoodEstimation :=
 ObservedData ∧ StatisticalModel
 → ConstructLikelihood
 → MaximiseOverParameterSpace

MaximumLikelihoodEstimator :=
 θ_hat_MLE=argmax_{θ∈Θ}L(θ|X)

ScoreFunction := U(θ)=∂ℓ(θ)/∂θ
FisherInformation := I(θ)=E_θ[U(θ)U(θ)^T]
ObservedInformation := J(θ)=-∇²ℓ(θ)

MLEAsymptoticNormality :=
 sqrt(n)(θ_hat_MLE-θ0)
 →N(0,I_1(θ0)^(-1))
 under regularity conditions

MethodOfMoments :=
 SampleMoments=ModelMoments
 → SolveForParameter

LeastSquaresEstimation :=
 Parameter → MinimiseSquaredResiduals

OrdinaryLeastSquares :=
 β_hat=argmin_β||y-Xβ||²

BayesianEstimation :=
 Prior ∧ Likelihood → Posterior → BayesEstimator

PriorDistribution := p(θ)
PosteriorDistribution := p(θ|x)∝p(x|θ)p(θ)
PosteriorMeanEstimator := E[θ|x] under squared-error loss
PosteriorMedianEstimator := Median(θ|x) under absolute-error loss
MAPEstimator := argmax_θp(θ|x)

LossFunction := L(θ,a)
RiskFunction := R(θ,δ)=E_θ[L(θ,δ(X))]
SquaredErrorLoss := (a-θ)^2
AbsoluteErrorLoss := |a-θ|
MinimaxEstimator := Estimator minimising maximum risk over parameter space
BayesEstimator := Estimator minimising posterior expected loss

RobustEstimator :=
 Estimator designed_to reduce sensitivity to contamination_or_model deviations

MEstimator := argmin_θΣ_iρ(x_i,θ)
InfluenceFunction := LocalContaminationSensitivityOfEstimatorFunctional
BreakdownPoint := ContaminationFractionProducingArbitrarilySevereFailure

ShrinkageEstimator :=
 RawEstimate → ShrinkTowardTargetOrStructure → PotentialMSEImprovement

RegularisedEstimator :=
 DataFitObjective+Penalty → Estimate

RidgeEstimator := argmin_β{||y-Xβ||²+λ||β||²_2}
LassoEstimator := argmin_β{||y-Xβ||²+λ||β||_1}

BootstrapEstimation :=
 ObservedSample → Resample → RecomputeEstimator → ApproximateSamplingDistribution

BootstrapStandardError := SD(T*_1,...,T*_B)
BootstrapBias := Mean(T*)-T_observed
JackknifeEstimation := LeaveOneOutSamples → EstimatorReplicates → BiasVarianceApproximation
DeltaMethod := AsymptoticEstimatorDistribution ∧ SmoothTransformation → ApproximateTransformedDistribution

ConfidenceLevel := 1-α
CoverageProbability := P_θ(L(X)≤θ≤U(X))
WaldInterval := θ_hat±z_(1-α/2)SE_hat(θ_hat)
TInterval := Estimate±t_(1-α/2,ν)EstimatedSE
LikelihoodRatioInterval := ParameterValuesSatisfyingLikelihoodRatioCriterion
ProfileLikelihood := TargetParameter → OptimiseNuisanceParameters → ProfileObjective

NuisanceParameter := ModelParameter required_for model but not primary estimand
Identifiability := DistinctParameterValues induce distinct observable distributions
NonIdentifiableParameter := MultipleParameterValues imply same observable distribution
IdentificationConstraint := ValidEstimation requires identifiable target or explicit partial-identification framework
ParameterBoundary := Parameter on boundary of admissible space
BoundaryInference := MayRequireNonstandardAsymptoticsOrConstrainedMethods

SamplingDesign := MechanismDeterminingSampleSelection
DesignBasedEstimation := SamplingDesignDefinesRandomnessForEstimatorProperties
ModelBasedEstimation := StatisticalModelDefinesStochasticAssumptionsForEstimatorProperties

SurveyEstimation :=
 SamplingWeights ∧ InclusionProbabilities ∧ Stratification ∧ Clustering
 → DesignAwareEstimate

HorvitzThompsonEstimator :=
 T_hat=Σ_{i∈s}y_i/π_i

WeightedEstimator := Observations ∧ Weights → WeightedEstimate

MissingDataEstimation :=
 MissingnessMechanism ∧ ObservedData ∧ EstimationProcedure
 → Estimate

CompleteCaseEstimation := Estimator applied_to complete eligible records

MultipleImputation :=
 GenerateMultipleCompletedDatasets
 → EstimateEach
 → CombineEstimatesAndUncertainty

InverseProbabilityWeighting :=
 ObservationWeight=InverseEstimatedSelectionOrResponseProbability

EstimatorUncertainty :=
 SamplingUncertainty ∨ PosteriorUncertainty ∨ ModelUncertainty ∨ MeasurementUncertainty

ModelMisspecification :=
 AssumedModel differs_from data-generating process

SandwichVariance :=
 RobustAsymptoticVariance under estimating-equation framework

PredictionEstimationDistinction :=
 Estimation targets parameter_or_functional
 whereas Prediction targets unobserved outcome

EstimatorProperties := {
 Bias,Variance,StandardError,MeanSquaredError,Consistency,Efficiency,Sufficiency,Robustness
}

EstimatorSelection :=
 CandidateEstimators
 → CompareEstimatorProperties
 → MatchToTargetModelDesignAndLoss
 → SelectProcedure

StatisticalEstimationConstraint :=
 ConstraintOn(
  Population,Sample,TargetParameter,Estimand,ParameterSpace,
  StatisticalModel,SamplingDesign,Estimator,Assumptions
 )

StatisticalEstimationConsistency :=
 Target ∧ Model ∧ Data ∧ Estimator ∧ SamplingFramework ∧ UncertaintyMethod
 jointly_compatible

InvalidStatisticalEstimation :=
 UndefinedTarget
 ∨ NonIdentifiableParameter
 ∨ InvalidEstimatorDomain
 ∨ IncompatibleDataAndModel
 ∨ InvalidSamplingAssumption
 ∨ FailedOptimisation
 ∨ IncorrectUncertaintyMethod

StatisticalEstimationValidation :=
 ResolvePopulation
 → ResolveSampleAndData
 → ResolveTargetParameterOrEstimand
 → ResolveParameterSpace
 → ResolveStatisticalModel
 → ResolveSamplingDesign
 → ValidateIdentifiability
 → ResolveEstimator
 → ComputeEstimate
 → ResolveSamplingOrPosteriorDistribution
 → QuantifyUncertainty
 → EvaluateEstimatorProperties
 → ValidateAssumptions
 → ValidateResult

StatisticalEstimationState := {
 Population,Sample,Data,TargetParameter,Estimand,ParameterSpace,StatisticalModel,
 SamplingDesign,Estimator,Estimate,SamplingDistribution,PosteriorDistribution,
 Bias,Variance,StandardError,MeanSquaredError,IntervalEstimate,Assumptions,
 Evidence,Confidence,Provenance
}

SEMANTIC_EQUIVALENCE(EstimationA,EstimationB) :=
 same(TargetParameterOrEstimand)
 ∧ same(DataSemantics)
 ∧ same(StatisticalModel)
 ∧ same(SamplingDesign)
 ∧ same(EstimatorDefinition)
 ∧ same(UncertaintyFramework)

CoreClasses := {
 StatisticalEstimation,Population,Sample,StatisticalModel,Parameter,ParameterSpace,
 Estimand,Estimator,Estimate,PointEstimation,IntervalEstimation,ConfidenceInterval,
 SamplingDistribution,Bias,Variance,StandardError,MeanSquaredError,
 MaximumLikelihoodEstimator,BayesianEstimator,RobustEstimator,SamplingDesign
}

CoreRelations := {
 estimates,targets,computedFrom,sampledFrom,assumesModel,belongsToParameterSpace,
 hasSamplingDistribution,hasPosteriorDistribution,hasBias,hasVariance,
 hasStandardError,hasMSE,constrainedBy,equivalentTo,tracesTo
}

TypedRelations :=
 Estimator --estimates--> Parameter
 Estimator --targets--> Estimand
 Estimate --computedFrom--> Sample
 Sample --sampledFrom--> Population
 Estimator --assumesModel--> StatisticalModel
 Parameter --belongsToParameterSpace--> ParameterSpace
 Estimator --hasSamplingDistribution--> SamplingDistribution
 BayesianEstimator --hasPosteriorDistribution--> PosteriorDistribution
 Estimator --hasBias--> Bias
 Estimator --hasVariance--> Variance
 Estimator --hasStandardError--> StandardError
 Estimator --hasMSE--> MeanSquaredError
 Estimate --tracesTo--> Provenance

InferenceRules := {
 Estimator(T) ∧ ObservedData(x) ⇒ Estimate=T(x),
 UnbiasedEstimator(T,θ) ⇒ E_θ[T]=θ,
 Bias(T)=0 ⇒ MSE(T)=Var(T),
 SamplingVariance(T) ⇒ StandardError(T)=sqrt(SamplingVariance(T)),
 ConsistentEstimator(T_n,θ) ⇒ T_n→θ under specified convergence,
 Likelihood ∧ IdentifiableParameter ⇒ MaximumLikelihoodEstimationEligible,
 Prior ∧ Likelihood ⇒ Posterior,
 NonIdentifiableParameter ⇒ FlagEstimationValidityRisk
}

CanonicalStatisticalEstimationRepresentation :=
 ParseEstimationRequest
 → ResolvePopulation
 → ResolveSampleAndData
 → ResolveTargetParameterOrEstimand
 → ResolveParameterSpace
 → ResolveStatisticalModel
 → ResolveSamplingDesign
 → ValidateIdentifiability
 → ResolveEstimator
 → ComputeEstimate
 → ResolveSamplingDistributionOrPosterior
 → QuantifyUncertainty
 → EvaluateBiasVarianceMSEConsistencyEfficiencyRobustness
 → ValidateEstimation

EstimationGraph := G=(V,E)

V := {
 Populations,Samples,StatisticalModels,Parameters,ParameterSpaces,Estimands,
 Estimators,Estimates,SamplingDistributions,PosteriorDistributions,Biases,
 Variances,StandardErrors,MeanSquaredErrors,ConfidenceIntervals,
 SamplingDesigns,Evidence,ProvenanceNodes
}

E := {
 estimates,targets,computedFrom,sampledFrom,assumesModel,belongsToParameterSpace,
 hasSamplingDistribution,hasPosteriorDistribution,hasBias,hasVariance,
 hasStandardError,hasMSE,constrainedBy,equivalentTo,tracesTo
}

CONFIDENCE(StatisticalEstimationResolution) :=
 weighted_mean(
  TargetResolution,
  PopulationSampleResolution,
  ModelResolution,
  SamplingDesignResolution,
  IdentifiabilityValidity,
  EstimatorValidity,
  UncertaintyResolution,
  ProvenanceQuality
 )

AI_QUERY_INTERPRETATION :=
 Query
 → DetectStatisticalEstimationIntent
 → IdentifyPopulationSampleAndTarget
 → ResolveStatisticalModelAndSamplingDesign
 → IdentifyPointIntervalLikelihoodBayesianOrRobustEstimation
 → ResolveRequestedEstimateUncertaintyAndEstimatorProperties

AI_REASONING_FLOW :=
 Query
 → ParseStatisticalEstimationProblem
 → ResolvePopulation
 → ResolveSampleAndObservedData
 → ResolveTargetParameterOrEstimand
 → ResolveParameterSpace
 → ResolveStatisticalModel
 → ResolveSamplingDesign
 → ValidateIdentifiability
 → SelectEstimator
 → ComputeEstimate
 → ResolveSamplingDistributionOrPosterior
 → ComputeBiasVarianceStandardErrorAndMSEAsRequired
 → ConstructIntervalEstimateIfRequested
 → EvaluateConsistencyEfficiencyAndRobustness
 → ValidateAssumptions
 → UpdateConfidence
 → PreserveProvenance
 → GenerateExplanation

AI_CONSUMPTION := {
 ParseStatisticalEstimationEntities,ParseTypedRelations,ResolvePopulations,
 ResolveSamples,ResolveObservedData,ResolveStatisticalModels,ResolveSamplingDesigns,
 ResolveParameters,ResolveEstimands,ResolveParameterSpaces,ValidateIdentifiability,
 ResolveEstimators,ComputePointEstimates,ComputeIntervalEstimates,
 ComputeMaximumLikelihoodEstimates,ComputeMethodOfMomentsEstimates,
 ComputeLeastSquaresEstimates,ComputeBayesianEstimates,ComputeRobustEstimates,
 ResolveSamplingDistributions,ResolvePosteriorDistributions,ComputeBias,
 ComputeVariance,ComputeStandardErrors,ComputeMeanSquaredErrors,EvaluateConsistency,
 EvaluateEfficiency,EvaluateSufficiency,EvaluateRobustness,ResolveBootstrapMethods,
 ResolveMissingDataMethods,ValidateStatisticalEstimation,CompareSemanticEquivalence,
 UpdateConfidence,PreserveProvenance,GenerateFormalExplanation
}

UPDATE_MODEL :=
 NewDataModelTargetEstimatorDesignOrEvidence
 → ValidateInput
 → IdentifyAffectedEstimationClaims
 → UpdateStatisticalEstimationState
 → RevalidateIdentifiability
 → RecomputeEstimate
 → RecomputeSamplingOrPosteriorDistribution
 → RecomputeUncertainty
 → ReevaluateEstimatorProperties
 → RevalidateAssumptions
 → RecomputeConfidence
 → PreserveProvenance
 → VersionModel

DataUpdate :=
 ChangedObservedData
 ⇒ recompute(Estimate,SamplingDistributionApproximation,StandardError,IntervalEstimate)

ModelUpdate :=
 ChangedStatisticalModel
 ⇒ reevaluate(Identifiability,EstimatorDefinition,SamplingDistribution,Assumptions)

EstimatorUpdate :=
 ChangedEstimator
 ⇒ recompute(Estimate,Bias,Variance,StandardError,MeanSquaredError,Efficiency,Robustness)

FLOW_HASH_INPUT :=
 canonical(
  ONTOLOGY,
  CoreClasses,
  CoreRelations,
  InferenceRules,
  StatisticalEstimationOntology,
  StatisticalEstimationValidation,
  StatisticalEstimationConstraint,
  SEMANTIC_EQUIVALENCE,
  AI_REASONING_FLOW
 )

FLOW_HASH := HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {
 StatisticalEstimationOntologyResolution,
 Population,
 Sample,
 StatisticalModel,
 SamplingDesign,
 TargetParameter,
 Estimand,
 ParameterSpace,
 Estimator,
 Estimate,
 SamplingDistribution,
 PosteriorDistribution,
 Bias,
 Variance,
 StandardError,
 MeanSquaredError,
 IntervalEstimate,
 IdentifiabilityStatus,
 EstimatorPropertyStatus,
 ValidationStatus,
 Confidence,
 Provenance,
 Explanation
}

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA