Point Estimation Ontology

FULL EXECUTABLE FLOW ALGEBRA — POINT ESTIMATION ONTOLOGY
Kireth Research — Semantic Library
URL: https://kirethresearch.org/booksai-core/ontologies/probability-statistics-ontologies/statistical-estimation/point-estimation-ontology.html

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

BEGIN_FLOW

PointEstimation :=
 Data ∧ TargetParameter ∧ Estimator ∧ RealisedEstimate ∧ EstimatorProperties ∧ UncertaintyContext

Population := TargetCollectionOrGeneratingProcess
Sample := {X_1,...,X_n}
ObservedSample := {x_1,...,x_n}
Parameter := θ
ParameterSpace := Θ
Estimand := PreciselyDefinedTargetQuantity
PointEstimator := T(X_1,...,X_n)
PointEstimate := θ_hat=T(x_1,...,x_n)

EstimatorVsEstimate :=
 Estimator=RandomRuleBeforeObservation
 whereas Estimate=RealisedValueAfterObservation

SamplingDistribution := DistributionOf(T(X_1,...,X_n))
EstimatorExpectation := E_θ[T]
EstimatorVariance := Var_θ(T)
EstimatorStandardError := SE(T)=sqrt(Var_θ(T))
Bias := Bias_θ(T)=E_θ[T]-θ
UnbiasedEstimator := E_θ[T]=θ
Consistency := T_n→θ in probability as n→∞
StrongConsistency := T_n→θ almost surely
AsymptoticUnbiasedness := Bias_θ(T_n)→0

MeanSquaredError := MSE_θ(T)=E_θ[(T-θ)^2]
BiasVarianceDecomposition := MSE_θ(T)=Var_θ(T)+Bias_θ(T)^2
RootMeanSquaredError := RMSE=sqrt(MSE)

Efficiency := RelativeEstimatorPerformance under specified variance_or_loss criterion
MinimumVarianceUnbiasedEstimator := UnbiasedEstimator with smallest variance in specified class
SufficientStatistic := Statistic retaining all sample information about θ under formal criterion
CompleteStatistic := E_θ[g(S)]=0 for all θ ⇒ g(S)=0 almost surely

RaoBlackwellImprovement :=
 T → E[T|S]
 where S sufficient
 → NoGreaterSquaredErrorRisk
 under applicable conditions

LehmannScheffeRelation :=
 CompleteSufficientStatistic ∧ UnbiasedEstimator
 ⇒ UniqueUMVUE
 under theorem conditions

Likelihood := L(θ|x)=f(x|θ)
LogLikelihood := ℓ(θ)=logL(θ|x)
MaximumLikelihoodPointEstimator := θ_hat_MLE=argmax_{θ∈Θ}L(θ|x)
ScoreEquation := ∂ℓ(θ)/∂θ=0

MLEProcedure :=
 SpecifyModel → ConstructLikelihood → DefineParameterSpace
 → OptimiseLikelihood → ValidateOptimum → ReturnPointEstimate

MethodOfMomentsEstimator :=
 SampleMoments=TheoreticalMoments
 → SolveForθ

LeastSquaresPointEstimator :=
 θ_hat=argmin_θ Σ_i Residual_i(θ)^2

BayesianPointEstimator :=
 PosteriorDistribution ∧ LossFunction
 → BayesAction

PosteriorMean := E[θ|x] under squared-error loss
PosteriorMedian := Median(θ|x) under absolute-error loss
PosteriorMode := argmax_θ p(θ|x)
MAPEstimator := PosteriorMode

SampleMeanEstimator := X_bar=(1/n)Σ_iX_i
PopulationMeanTarget := μ=E[X]
MeanPointEstimate := x_bar=(1/n)Σ_i x_i
SampleMeanUnbiasedness := E[X_bar]=μ under finite-mean compatible sampling
SampleMeanVariance := IIDSample ∧ Var(X)=σ² ⇒ Var(X_bar)=σ²/n

SampleProportionEstimator := p_hat=X/n
PopulationProportionTarget := p
ProportionUnbiasedness := E[p_hat]=p under Bernoulli sampling
ProportionVariance := Var(p_hat)=p(1-p)/n

SampleVarianceEstimator := S²=[1/(n-1)]Σ_i(X_i-X_bar)^2
PopulationVarianceTarget := σ²
SampleVarianceUnbiasedness := E[S²]=σ² under iid finite-variance sampling

SampleMedianEstimator := Median(X_1,...,X_n)
PopulationMedianTarget := m
SampleQuantileEstimator := Q_hat(q)
PopulationQuantileTarget := Q(q)

DifferenceInMeansEstimator := X_bar_1-X_bar_2
DifferenceInProportionsEstimator := p_hat_1-p_hat_2
PlugInEstimator := θ=T(F) ⇒ θ_hat=T(F_n)
EmpiricalDistribution := F_n(x)=(1/n)Σ_i1_(X_i≤x)

ShrinkageEstimator :=
 RawPointEstimate → ShrinkTowardTarget → PotentialMSEImprovement

RegularisedPointEstimator :=
 DataFitObjective + Penalty → PointEstimate

RobustPointEstimator :=
 Estimator designed_to reduce sensitivity_to contamination

TrimmedMeanEstimator :=
 RemoveSpecifiedTailFraction → ComputeMean

MEstimator :=
 argmin_θ Σ_iρ(X_i,θ)

InfluenceFunction := LocalContaminationSensitivityOfEstimatorFunctional
BreakdownPoint := ContaminationFractionRequiredForArbitrarilyLargeFailure

EstimatorEquivariance := {
 TranslationEquivariance,
 ScaleEquivariance,
 AffineEquivariance,
 TransformationInvariance
}

TranslationEquivariance := T(X+c)=T(X)+c
ScaleEquivariance := T(aX)=aT(X) under compatible parameter semantics

EstimatorRisk := R(θ,T)=E_θ[L(θ,T(X))]
SquaredErrorRisk := E_θ[(T-θ)^2]=MSE
AbsoluteErrorRisk := E_θ[|T-θ|]

MinimaxPointEstimator :=
 Estimator minimising maximum risk over Θ

BayesPointEstimator :=
 Estimator minimising posterior expected loss

SamplingError := PointEstimate-TrueParameter
StandardError := StandardDeviationOfSamplingDistribution
PointEstimateDoesNotEqualTruth := θ_hat estimates θ ↛ θ_hat=θ
Precision := InverseRelationToSamplingVariability
HigherPrecision := SmallerStandardError under same target_and_scale
BiasPrecisionDistinction := LowBias and HighPrecision are distinct properties

AsymptoticNormality :=
 sqrt(n)(T_n-θ) → N(0,V(θ))
 under regularity conditions

AsymptoticVariance := V(θ)/n
AsymptoticStandardError := sqrt(V_hat(θ)/n)

DeltaMethodPointTransformation :=
 T_n estimates θ ∧ g smooth
 ⇒ g(T_n) estimates g(θ)

MLEInvariance :=
 θ_hat_MLE estimates θ
 ⇒ g(θ_hat_MLE) estimates g(θ)
 under applicable MLE transformation conditions

BootstrapPointEstimation :=
 ObservedSample → BootstrapReplicates → PointEstimatorDistributionApproximation

BootstrapBiasCorrection :=
 θ_hat_corrected=θ_hat-EstimatedBootstrapBias

MissingDataPointEstimation :=
 MissingnessMechanism ∧ ObservedData ∧ EstimationMethod
 → PointEstimate

MultipleImputationPointEstimate :=
 MultipleCompletedDatasets
 → SeparatePointEstimates
 → CombineUnderSpecifiedRules

WeightedPointEstimator :=
 Observations ∧ Weights → WeightedEstimate

WeightedMeanEstimator :=
 Σ_iw_iX_i/Σ_iw_i

SurveyPointEstimation :=
 SamplingDesign ∧ InclusionProbabilities ∧ Weights
 → DesignBasedPointEstimate

HorvitzThompsonTotalEstimator :=
 T_hat_HT=Σ_{i∈s}y_i/π_i

ModelBasedEstimation :=
 StatisticalModel ∧ Data → ParameterEstimate

DesignBasedEstimation :=
 RandomnessFromSamplingDesign → PopulationQuantityEstimate

FrequentistPointEstimation :=
 ParameterFixed ∧ DataRandom ∧ EstimatorSamplingProperties

BayesianPointEstimation :=
 PosteriorDistribution → DecisionBasedPointSummary
EstimatorSelection :=
 CandidateEstimators
 → CompareProperties
 → SelectEstimator
 according_to target, assumptions, loss, and robustness needs

EstimatorComparisonCriteria := {
 Bias,
 Variance,
 MeanSquaredError,
 Consistency,
 Efficiency,
 Robustness,
 Sufficiency,
 ComputationalCost
}

PointEstimationConstraint :=
 ConstraintOn(
  TargetParameter,
  ParameterSpace,
  Data,
  StatisticalModel,
  SamplingDesign,
  EstimatorDefinition,
  LossFunction
 )

PointEstimationConsistency :=
 TargetParameter
 ∧ EstimatorDefinition
 ∧ DataScope
 ∧ Model
 ∧ SamplingDesign
 jointly_compatible

InvalidPointEstimation :=
 UndefinedTarget
 ∨ NonIdentifiableParameter
 ∨ EstimatorOutsideDomain
 ∨ InvalidDataScope
 ∨ ViolatedCriticalAssumption
 ∨ FailedOptimisation
 ∨ UnresolvedWeighting

PointEstimationValidation :=
 ResolvePopulation
 → ResolveSample
 → ResolveTargetParameter
 → ResolveParameterSpace
 → ResolveStatisticalModel
 → ValidateIdentifiability
 → ResolveEstimator
 → ValidateEstimatorDomain
 → ComputePointEstimate
 → ResolveSamplingProperties
 → EvaluateBiasVarianceMSE
 → ValidateResult

PointEstimationState := {
 Population,
 Sample,
 Data,
 Parameter,
 ParameterSpace,
 Estimand,
 StatisticalModel,
 Estimator,
 PointEstimate,
 Bias,
 Variance,
 StandardError,
 MeanSquaredError,
 ConsistencyStatus,
 EfficiencyStatus,
 RobustnessStatus,
 Evidence,
 Confidence,
 Provenance
}

SEMANTIC_EQUIVALENCE(PointEstimateA,PointEstimateB) :=
 same(TargetParameterOrEstimand)
 ∧ same(DataScope)
 ∧ same(ModelSemantics)
 ∧ same(EstimatorDefinition)
 ∧ same(WeightingAndSamplingFramework)

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

CoreClasses := {
 PointEstimation,
 PointEstimator,
 PointEstimate,
 Parameter,
 Estimand,
 ParameterSpace,
 SamplingDistribution,
 Bias,
 Variance,
 StandardError,
 MeanSquaredError,
 MaximumLikelihoodEstimator,
 MethodOfMomentsEstimator,
 LeastSquaresEstimator,
 BayesianPointEstimator,
 RobustPointEstimator,
 WeightedPointEstimator,
 StatisticalModel,
 Sample,
 Population
}

CoreRelations := {
 estimates,
 targets,
 computedFrom,
 assumesModel,
 belongsToParameterSpace,
 hasSamplingDistribution,
 hasBias,
 hasVariance,
 hasStandardError,
 hasMSE,
 minimises,
 maximises,
 weightedBy,
 constrainedBy,
 equivalentTo,
 tracesTo
}

TypedRelations :=
 PointEstimator --estimates--> Parameter
 PointEstimator --targets--> Estimand
 PointEstimate --computedFrom--> Sample
 PointEstimator --assumesModel--> StatisticalModel
 Parameter --belongsToParameterSpace--> ParameterSpace
 PointEstimator --hasSamplingDistribution--> SamplingDistribution
 PointEstimator --hasBias--> Bias
 PointEstimator --hasVariance--> Variance
 PointEstimator --hasStandardError--> StandardError
 PointEstimator --hasMSE--> MeanSquaredError
 LeastSquaresEstimator --minimises--> SquaredErrorObjective
 MaximumLikelihoodEstimator --maximises--> Likelihood
 WeightedPointEstimator --weightedBy--> Weight
 PointEstimate --tracesTo--> Provenance

InferenceRules := {
 Estimator(T) ∧ ObservedSample(x)
  ⇒ PointEstimate=T(x),

 UnbiasedEstimator(T,θ)
  ⇒ E_θ[T]=θ,

 Bias(T)=0
  ⇒ MSE(T)=Var(T),

 SamplingVariance(T)
  ⇒ SE(T)=sqrt(SamplingVariance(T)),

 ConsistentEstimator(T_n,θ)
  ⇒ T_n→θ under specified convergence,

 LikelihoodAvailable ∧ ParameterIdentifiable
  ⇒ CandidateMLE=argmaxLikelihood,

 NonIdentifiableParameter
  ⇒ FlagPointEstimationInvalid
}

CanonicalPointEstimationRepresentation :=
 ParsePointEstimationRequest
 → ResolvePopulation
 → ResolveSampleAndData
 → ResolveTargetParameterOrEstimand
 → ResolveParameterSpace
 → ResolveStatisticalModel
 → ValidateIdentifiability
 → ResolveEstimator
 → ComputePointEstimate
 → ResolveSamplingDistribution
 → ComputeBiasVarianceStandardErrorAndMSEAsRequired
 → EvaluateEstimatorProperties
 → ValidatePointEstimate

PointEstimationGraph := G=(V,E)

V := {
 PointEstimations,
 PointEstimators,
 PointEstimates,
 Parameters,
 Estimands,
 ParameterSpaces,
 Populations,
 Samples,
 StatisticalModels,
 SamplingDistributions,
 Biases,
 Variances,
 StandardErrors,
 MeanSquaredErrors,
 Weights,
 Evidence,
 ProvenanceNodes
}

E := {
 estimates,
 targets,
 computedFrom,
 assumesModel,
 belongsToParameterSpace,
 hasSamplingDistribution,
 hasBias,
 hasVariance,
 hasStandardError,
 hasMSE,
 minimises,
 maximises,
 weightedBy,
 constrainedBy,
 equivalentTo,
 tracesTo
}

CONFIDENCE(PointEstimationResolution) :=
 weighted_mean(
  TargetResolution,
  PopulationSampleResolution,
  ModelResolution,
  IdentifiabilityValidity,
  EstimatorValidity,
  SamplingPropertyResolution,
  RobustnessAssessment,
  ProvenanceQuality
 )

AI_QUERY_INTERPRETATION :=
 Query
 → DetectPointEstimationIntent
 → IdentifyPopulationSampleAndData
 → IdentifyTargetParameterOrEstimand
 → ResolveStatisticalModel
 → IdentifyEstimatorType
 → IdentifyRequestedEstimateOrEstimatorProperties

AI_REASONING_FLOW :=
 Query
 → ParsePointEstimationProblem
 → ResolvePopulation
 → ResolveSampleAndData
 → ResolveTargetParameterOrEstimand
 → ResolveParameterSpace
 → ResolveStatisticalModel
 → ValidateIdentifiability
 → SelectOrResolvePointEstimator
 → ComputePointEstimate
 → ResolveSamplingDistributionIfRequired
 → ComputeBiasVarianceStandardErrorAndMSEIfRequired
 → EvaluateConsistencyEfficiencyAndRobustness
 → ValidateAssumptions
 → UpdateConfidence
 → PreserveProvenance
 → GenerateExplanation
AI_CONSUMPTION := {
 ParsePointEstimationEntities,
 ParseTypedRelations,
 ResolvePopulations,
 ResolveSamples,
 ResolveData,
 ResolveParameters,
 ResolveEstimands,
 ResolveParameterSpaces,
 ResolveStatisticalModels,
 ValidateIdentifiability,
 ResolvePointEstimators,
 ComputePointEstimates,
 ComputeSampleMeanEstimates,
 ComputeProportionEstimates,
 ComputeVarianceEstimates,
 ComputeQuantileEstimates,
 ComputeMaximumLikelihoodEstimates,
 ComputeMethodOfMomentsEstimates,
 ComputeLeastSquaresEstimates,
 ComputeBayesianPointEstimates,
 ComputeWeightedPointEstimates,
 ResolveSamplingDistributions,
 ComputeBias,
 ComputeVariance,
 ComputeStandardErrors,
 ComputeMeanSquaredErrors,
 EvaluateConsistency,
 EvaluateEfficiency,
 EvaluateRobustness,
 ValidatePointEstimation,
 CompareSemanticEquivalence,
 UpdateConfidence,
 PreserveProvenance,
 GenerateFormalExplanation
}

UPDATE_MODEL :=
 NewDataModelTargetEstimatorOrEvidence
 → ValidateInput
 → IdentifyAffectedPointEstimationClaims
 → UpdatePointEstimationState
 → RevalidateIdentifiability
 → RecomputePointEstimate
 → RecomputeSamplingDistributionIfRequired
 → RecomputeBiasVarianceStandardErrorAndMSE
 → ReevaluateConsistencyEfficiencyAndRobustness
 → RecomputeConfidence
 → PreserveProvenance
 → VersionModel

DataUpdate :=
 ChangedSampleData
 ⇒ recompute(
  PointEstimate,
  BiasEstimate,
  VarianceEstimate,
  StandardError,
  MeanSquaredError
 )

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

EstimatorUpdate :=
 ChangedPointEstimator
 ⇒ recompute(
  PointEstimate,
  Bias,
  Variance,
  MeanSquaredError,
  Efficiency,
  Robustness
 )

FLOW_HASH_INPUT :=
 canonical(
  ONTOLOGY,
  CoreClasses,
  CoreRelations,
  InferenceRules,
  PointEstimation,
  PointEstimationValidation,
  PointEstimationConstraint,
  SEMANTIC_EQUIVALENCE,
  AI_REASONING_FLOW
 )

FLOW_HASH :=
 HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {
 PointEstimationEntityResolution,
 Population,
 Sample,
 StatisticalModel,
 TargetParameter,
 Estimand,
 ParameterSpace,
 PointEstimator,
 PointEstimate,
 SamplingDistribution,
 Bias,
 Variance,
 StandardError,
 MeanSquaredError,
 ConsistencyStatus,
 EfficiencyStatus,
 RobustnessStatus,
 IdentifiabilityStatus,
 ValidationStatus,
 Confidence,
 Provenance,
 Explanation
}

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA