Estimator Bias Ontology

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

BEGIN_FLOW

EstimatorBias := EstimatorProperty ∧ SystematicExpectedDeviation ∧ DifferenceBetweenExpectedEstimateAndTarget
Population := TargetCollectionOrGeneratingProcess
Sample := {X_1,...,X_n}
Parameter := θ
ParameterSpace := Θ
Estimand := τ(θ)
Estimator := T(X_1,...,X_n)
Estimate := T(x_1,...,x_n)
EstimatorExpectation := E_θ[T]

BiasDefinition := Bias_θ(T)=E_θ[T]-τ(θ)
ParameterEstimatorBias := Bias_θ(T)=E_θ[T]-θ
PositiveBias := Bias_θ(T)>0
NegativeBias := Bias_θ(T)<0
ZeroBias := Bias_θ(T)=0
UnbiasedEstimator := Bias_θ(T)=0 for all θ∈Θ
BiasedEstimator := Existsθ∈Θ:Bias_θ(T)≠0
AbsoluteBias := |Bias_θ(T)|
RelativeBias := Bias_θ(T)/τ(θ) where denominator meaningful and nonzero
PercentageBias := 100×RelativeBias

EstimatorError := T-τ(θ)
ExpectedEstimatorError := E_θ[T-τ(θ)]=Bias_θ(T)
BiasVsRealisedError := Bias is expected systematic error whereas realised error is sample-specific

Variance := Var_θ(T)
MeanSquaredError := MSE_θ(T)=E_θ[(T-τ(θ))²]
BiasVarianceDecomposition := MSE_θ(T)=Var_θ(T)+Bias_θ(T)²
RootMeanSquaredError := RMSE=sqrt(MSE)
UnbiasedMSERelation := Bias_θ(T)=0 ⇒ MSE_θ(T)=Var_θ(T)
BiasVarianceTradeoff := ControlledBias may reduce variance enough_to lower MSE

FiniteSampleBias := BiasPresentAtFiniteSampleSize
BiasAsFunctionOfSampleSize := b_n(θ)=E_θ[T_n]-τ(θ)
AsymptoticUnbiasedness := lim_{n→∞}Bias_θ(T_n)=0
ConsistencyDoesNotRequireExactUnbiasedness := ConsistentEstimator may be biased at finite n
FirstOrderBias := Bias_n=O(1/n)

SampleMean := X_bar=(1/n)Σ_iX_i
SampleMeanBias := E[X_bar]-μ=0 under equal finite means
SampleProportion := p_hat=X/n
SampleProportionBias := E[p_hat]-p=0 under binomial sampling

SampleVarianceWithNMinus1 := S²=[1/(n-1)]Σ_i(X_i-X_bar)²
SampleVarianceUnbiasedness := E[S²]=σ² under iid finite-variance sampling
VarianceEstimatorWithN := V_n=[1/n]Σ_i(X_i-X_bar)²
VarianceEstimatorWithNBias := E[V_n]=[(n-1)/n]σ²
VarianceEstimatorWithNBiasAmount := Bias(V_n)=-σ²/n
BesselCorrection := ReplaceDenominatorNWithNMinus1 → RemoveVarianceBiasUnderIIDConditions

SampleStandardDeviationBias := E[S] generally ≠ σ
NonlinearTransformationBias := E[g(T)] generally ≠ g(E[T])
JensenBiasRelation := NonlinearTransformation may induce bias even when original estimator unbiased

MaximumLikelihoodEstimator := θ_hat_MLE
MLEFiniteSampleBias := MLE may be biased in finite samples
MLEAsymptoticBias := UnderRegularityConditions Bias(θ_hat_MLE) often tends_to zero
NormalVarianceMLE := σ²_hat_MLE=(1/n)Σ_i(X_i-X_bar)²
NormalVarianceMLEBias := Bias(σ²_hat_MLE)=-σ²/n

UniformUpperBoundMLE := X_(n)=max_iX_i for Uniform(0,θ)
UniformUpperBoundMLEExpectation := E[X_(n)]=nθ/(n+1)
UniformUpperBoundMLEBias := Bias(X_(n))=-θ/(n+1)
UniformUpperBoundUnbiasedCorrection := [(n+1)/n]X_(n)

ShrinkageBias := ShrinkageEstimator intentionally biased toward target_or_structure
RidgeBias := RidgeRegressionEstimator generally biased when λ>0
RegularisationBias := Penalty alters estimator expectation relative_to unpenalised target

BiasReduction := ModifyEstimator → ReduceExpectedDeviationFromTarget
AnalyticBiasCorrection := CorrectedEstimator=OriginalEstimator-EstimatedBias
BiasCorrectionRisk := BiasReduction may increase variance and need_not reduce MSE

JackknifeBiasEstimate := b_hat_J=(n-1)(T_bar_(.)-T)
JackknifeBiasCorrectedEstimator := T_J=T-b_hat_J
BootstrapBiasEstimate := b_hat_boot=Mean(T*)-T
BootstrapBiasCorrectedEstimator := T_BC=T-b_hat_boot

BootstrapBiasProcedure :=
 ObservedSample → GenerateBootstrapSamples → ComputeEstimatorReplicates
 → EstimateBootstrapExpectation → EstimateBias → OptionalBiasCorrection

SimulationBiasEstimate :=
 RepeatedSimulatedDatasets → ComputeEstimatorEachDataset → MeanEstimateMinusKnownTruth

SelectionBias := SamplingOrSelectionMechanism causes estimator expectation to differ_from target
NonresponseBias := ResponseMechanism related_to outcome_or covariates may bias naive estimator
MeasurementBias := SystematicMeasurementError propagates_to estimator bias
OmittedVariableBias := OmittedRelevantCorrelatedPredictor → CoefficientBias under standard conditions
EndogeneityBias := E[X^Tε]≠0 → OLSBiasOrInconsistency
ConfoundingBias := UncontrolledCommonCauses may distort causal estimand

BiasTerminologyConstraint :=
 EstimatorBias must_not be conflated_with social,cognitive,algorithmic,or dataset bias
 unless explicit relationship defined

ConditionalBias := E[T-τ|C]
ConditionalUnbiasedness := E[T|C]=τ under specified conditioning structure
DesignBias := ExpectationOverSamplingDesign minus finite-population target
ModelBias := ExpectationUnderStatisticalModel minus model estimand

BayesEstimatorBias := E_θ[δ_B(X)]-θ
PriorInducedBias := InformativePrior may pull posterior estimator toward prior location

BiasUnderMisspecification :=
 Estimator may converge_to pseudo-true target different_from intended scientific target

PseudoTrueParameter :=
 θ*=argmin_θ KL(TrueDistribution||ModelDistribution_θ)

MisspecificationBias := θ*-IntendedTarget under specified target mapping

BiasFromMissingData :=
 MissingnessMechanism ∧ IncompleteObservation ∧ NaiveAnalysis → PotentialBias

InverseProbabilityWeightingBias :=
 WeightModelMisspecification may induce bias

BiasAndStandardError :=
 Bias describes estimator centre displacement
 whereas StandardError describes estimator sampling spread

BiasAndPrecision := LowBias ↛ HighPrecision
BiasFunction := b(θ)=E_θ[T]-τ(θ)
UniformBiasBound := sup_{θ∈Θ}|Bias_θ(T)|

EstimatorBiasConstraint :=
 ConstraintOn(
  EstimatorDefinition,TargetEstimand,ProbabilityFramework,SampleSize,
  SamplingDesign,StatisticalModel,MissingnessMechanism
 )

EstimatorBiasConsistency :=
 Estimator ∧ Target ∧ ExpectationOperator ∧ DataGeneratingFramework ∧ SampleSize
 jointly_compatible

InvalidBiasClaim :=
 UndefinedEstimator
 ∨ UndefinedTarget
 ∨ UndefinedExpectation
 ∨ IncompatibleProbabilityFramework
 ∨ WrongSampleSizeContext
 ∨ ConflatedRealisedErrorAndExpectedBias

EstimatorBiasValidation :=
 ResolveEstimator
 → ResolveTargetParameterOrEstimand
 → ResolveProbabilityFramework
 → ResolveSampleSize
 → ResolveSamplingDesignAndModel
 → ComputeOrDeriveEstimatorExpectation
 → SubtractTarget
 → DetermineBiasDirectionAndMagnitude
 → ResolveFiniteSampleOrAsymptoticContext
 → ValidateResult

EstimatorBiasState := {
 Population,Sample,StatisticalModel,SamplingDesign,Parameter,Estimand,Estimator,
 SampleSize,EstimatorExpectation,BiasFunction,BiasValue,BiasDirection,AbsoluteBias,
 RelativeBias,Variance,MeanSquaredError,AsymptoticBias,BiasCorrection,
 Evidence,Confidence,Provenance
}

SEMANTIC_EQUIVALENCE(BiasA,BiasB) :=
 same(EstimatorDefinition)
 ∧ same(TargetEstimand)
 ∧ same(DataGeneratingFramework)
 ∧ same(SampleSizeContext)
 ∧ same(ExpectationOperator)
 ∧ same(ParameterisationOrEquivalentTransform)

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

CoreClasses := {
 EstimatorBias,PositiveBias,NegativeBias,ZeroBias,UnbiasedEstimator,BiasedEstimator,
 FiniteSampleBias,AsymptoticBias,RelativeBias,BiasCorrection,BiasCorrectedEstimator,
 BootstrapBias,JackknifeBias,SelectionBias,MeasurementBias,ModelBias,Estimator,
 Estimand,StatisticalModel
}

CoreRelations := {
 biasOf,targets,computedFrom,hasExpectation,hasDirection,hasMagnitude,contributesTo,
 correctedBy,dependsOn,evaluatedUnder,comparedWith,constrainedBy,equivalentTo,tracesTo
}

TypedRelations :=
 EstimatorBias --biasOf--> Estimator
 Estimator --targets--> Estimand
 EstimatorBias --computedFrom--> EstimatorExpectation
 Estimator --hasExpectation--> EstimatorExpectation
 EstimatorBias --hasDirection--> BiasDirection
 EstimatorBias --hasMagnitude--> AbsoluteBias
 EstimatorBias --contributesTo--> MeanSquaredError
 EstimatorBias --correctedBy--> BiasCorrection
 EstimatorBias --dependsOn--> SampleSize
 EstimatorBias --evaluatedUnder--> StatisticalModel
 EstimatorBias --comparedWith--> EstimatorVariance
 EstimatorBias --tracesTo--> Provenance

InferenceRules := {
 EstimatorExpectation ∧ TargetEstimand
  ⇒ Bias=EstimatorExpectation-TargetEstimand,

 Bias=0
  ⇒ EstimatorUnbiasedAtSpecifiedParameter,

 Bias=0ForAllParameters
  ⇒ UnbiasedEstimator,

 Bias>0
  ⇒ PositiveBias,

 Bias<0
  ⇒ NegativeBias,

 MSE
  = Variance+Bias²,

 AsymptoticBias→0
  ⇒ AsymptoticallyUnbiased,

 UndefinedExpectation
  ⇒ BiasUndefined
}

CanonicalEstimatorBiasRepresentation :=
 ParseEstimatorBiasRequest
 → ResolveEstimator
 → ResolveTargetParameterOrEstimand
 → ResolvePopulationSampleModelAndDesign
 → ResolveSampleSize
 → ResolveExpectationOperator
 → ComputeEstimatorExpectation
 → ComputeBias
 → ResolveDirectionMagnitudeAndRelativeBias
 → ComputeVarianceAndMSEIfRequested
 → ResolveBiasCorrectionIfRequested
 → ResolveAsymptoticBiasIfRequested
 → ValidateEstimatorBias

EstimatorBiasGraph := G=(V,E)

V := {
 EstimatorBiases,PositiveBiases,NegativeBiases,ZeroBiases,UnbiasedEstimators,
 BiasedEstimators,FiniteSampleBiases,AsymptoticBiases,RelativeBiases,
 BiasCorrections,BootstrapBiases,JackknifeBiases,SelectionBiases,
 MeasurementBiases,ModelBiases,Estimators,Estimands,StatisticalModels,
 SamplingDesigns,Evidence,ProvenanceNodes
}

E := {
 biasOf,targets,computedFrom,hasExpectation,hasDirection,hasMagnitude,contributesTo,
 correctedBy,dependsOn,evaluatedUnder,comparedWith,constrainedBy,equivalentTo,tracesTo
}

CONFIDENCE(EstimatorBiasResolution) :=
 weighted_mean(
  EstimatorResolution,
  TargetResolution,
  ProbabilityFrameworkResolution,
  ExpectationValidity,
  SampleSizeResolution,
  ModelAndDesignValidity,
  BiasCalculationValidity,
  ProvenanceQuality
 )

AI_QUERY_INTERPRETATION :=
 Query
 → DetectEstimatorBiasIntent
 → IdentifyEstimatorAndTarget
 → ResolveFiniteSampleOrAsymptoticContext
 → ResolveModelSamplingDesignAndExpectation
 → IdentifyRequestedBiasMagnitudeDirectionCorrectionOrMSERelation

AI_REASONING_FLOW :=
 Query
 → ParseEstimatorBiasProblem
 → ResolveEstimator
 → ResolveTargetParameterOrEstimand
 → ResolvePopulationSampleModelAndDesign
 → ResolveSampleSize
 → ResolveExpectationOperator
 → ComputeOrDeriveEstimatorExpectation
 → ComputeBias
 → DetermineBiasDirectionAndMagnitude
 → ComputeRelativeBiasIfMeaningful
 → ResolveVarianceAndMSERelationIfRequested
 → ResolveBiasCorrectionOrAsymptoticBiasIfRequested
 → ValidateBiasResult
 → UpdateConfidence
 → PreserveProvenance
 → GenerateExplanation

AI_CONSUMPTION := {
 ParseEstimatorBiasEntities,ParseTypedRelations,ResolveEstimators,ResolveEstimands,
 ResolveParameters,ResolvePopulations,ResolveSamples,ResolveStatisticalModels,
 ResolveSamplingDesigns,ResolveSampleSizes,ComputeEstimatorExpectations,ComputeBias,
 ComputeAbsoluteBias,ComputeRelativeBias,ComputeMeanSquaredError,
 ResolveBiasVarianceDecomposition,ResolveFiniteSampleBias,ResolveAsymptoticBias,
 ComputeBootstrapBias,ComputeJackknifeBias,ApplyBiasCorrections,
 DiagnoseSelectionBias,DiagnoseMeasurementBias,DiagnoseModelMisspecificationBias,
 ValidateEstimatorBiasSemantics,CompareSemanticEquivalence,UpdateConfidence,
 PreserveProvenance,GenerateFormalExplanation
}

UPDATE_MODEL :=
 NewDataModelTargetEstimatorSampleSizeOrEvidence
 → ValidateInput
 → IdentifyAffectedBiasClaims
 → UpdateEstimatorBiasState
 → RecomputeEstimatorExpectation
 → RecomputeBias
 → RecomputeRelativeBiasAndMSE
 → ReevaluateFiniteSampleAndAsymptoticBias
 → RecomputeBiasCorrectionIfRequired
 → RecomputeConfidence
 → PreserveProvenance
 → VersionModel

EstimatorUpdate :=
 ChangedEstimator
 ⇒ recompute(
  EstimatorExpectation,BiasValue,Variance,MeanSquaredError
 )

TargetUpdate :=
 ChangedEstimand
 ⇒ recompute(
  BiasValue,RelativeBias,SemanticEquivalence
 )

SampleSizeUpdate :=
 ChangedSampleSize
 ⇒ reevaluate(
  FiniteSampleBias,AsymptoticBias,Variance,MeanSquaredError
 )

FLOW_HASH_INPUT :=
 canonical(
  ONTOLOGY,
  CoreClasses,
  CoreRelations,
  InferenceRules,
  EstimatorBias,
  EstimatorBiasValidation,
  EstimatorBiasConstraint,
  SEMANTIC_EQUIVALENCE,
  AI_REASONING_FLOW
 )

FLOW_HASH := HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {
 EstimatorBiasEntityResolution,
 Population,
 Sample,
 StatisticalModel,
 SamplingDesign,
 Parameter,
 Estimand,
 Estimator,
 SampleSize,
 EstimatorExpectation,
 BiasValue,
 BiasDirection,
 AbsoluteBias,
 RelativeBias,
 Variance,
 MeanSquaredError,
 FiniteSampleBias,
 AsymptoticBias,
 BiasCorrection,
 ValidationStatus,
 Confidence,
 Provenance,
 Explanation
}

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA