Statistical Estimation

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

BEGIN_FLOW

StatisticalEstimation :=
 Data ∧ StatisticalModel ∧ TargetParameter ∧ Estimator ∧ Estimate ∧ UncertaintyQuantification ∧ Validation

Population := TargetCollectionOrGeneratingProcess
Sample := ObservedDataFromPopulationOrModel
Parameter := UnknownQuantityDescribingPopulationOrModel
ParameterSpace := Θ
Estimand := PreciselyDefinedTargetQuantity
Estimator := FunctionOfObservedData used_to estimate ParameterOrEstimand
EstimatorNotation := θ_hat=T(X_1,...,X_n)
Estimate := RealisedValueOfEstimator
PointEstimation := Data → Estimator → SingleEstimate
IntervalEstimation := Data → LowerBound ∧ UpperBound → IntervalEstimate
SamplingDistribution := DistributionOfEstimator under repeated sampling
EstimatorExpectation := E_θ[θ_hat]
EstimatorVariance := Var_θ(θ_hat)
EstimatorStandardError := SE(θ_hat)=sqrt(Var_θ(θ_hat))
Bias := Bias_θ(θ_hat)=E_θ[θ_hat]-θ
UnbiasedEstimator := E_θ[θ_hat]=θ
Consistency := θ_hat_n → θ in probability as n→∞
StrongConsistency := θ_hat_n → θ almost surely
MeanSquaredError := MSE_θ(θ_hat)=E_θ[(θ_hat-θ)^2]
BiasVarianceDecomposition := MSE=Var(θ_hat)+Bias(θ_hat)^2
RootMeanSquaredError := RMSE=sqrt(MSE)
Efficiency := RelativeEstimatorPerformance under specified variance_or_loss criterion
MinimumVarianceUnbiasedEstimator := UnbiasedEstimator with minimum variance in specified class

SufficientStatistic := Statistic retaining all sample information about parameter under formal criterion
FactorisationCriterion := Likelihood(Data|θ)=g(T(Data),θ)h(Data) ⇒ T may be sufficient under theorem conditions
CompleteStatistic := E_θ[g(T)]=0 for all θ ⇒ g(T)=0 almost surely under completeness definition
RaoBlackwellRelation := Estimator U ∧ SufficientStatistic T ⇒ E[U|T] has no greater squared-error risk under applicable conditions
LehmannScheffeRelation := CompleteSufficientStatistic ∧ UnbiasedFunctionOfStatistic ⇒ UniqueUMVUE under theorem conditions

Likelihood := L(θ|x)=f(x|θ)
LogLikelihood := ℓ(θ)=logL(θ|x)
MaximumLikelihoodEstimator := θ_hat_MLE=argmax_{θ∈Θ}L(θ|x)
ScoreFunction := U(θ)=∂ℓ(θ)/∂θ
ObservedInformation := J(θ)=-∂²ℓ(θ)/∂θ²
FisherInformation := I(θ)=E_θ[U(θ)^2] under regularity conditions
CramerRaoLowerBound := Var_θ(T)≥[g'(θ)]²/I_n(θ) under regularity conditions
MLEConsistency := θ_hat_MLE→θ0 under identifiability_and_regularity conditions
MLEAsymptoticNormality := sqrt(n)(θ_hat_MLE-θ0)→N(0,I_1(θ0)^(-1)) under regularity conditions
MLEStandardError := sqrt(diagonal(EstimatedInformationInverse))
MLEInvariance := η=g(θ) ⇒ η_hat_MLE=g(θ_hat_MLE) under applicable conditions

MethodOfMoments := SampleMoments=TheoreticalMoments → SolveForParameters
GeneralisedMethodOfMoments := MomentConditions → SampleAnalogues → MinimiseWeightedMomentDiscrepancy
LeastSquaresEstimation := Parameter → MinimiseSumOfSquaredResiduals
OrdinaryLeastSquares := β_hat=argmin_β||y-Xβ||²
OLSClosedForm := β_hat=(X^TX)^(-1)X^Ty when X^TX invertible
WeightedLeastSquares := β_hat_WLS=argmin_β(y-Xβ)^TW(y-Xβ)

RobustEstimation := EstimationProcedure designed_to reduce sensitivity to outliers_or_model deviations
MEstimator := θ_hat=argmin_θΣ_iρ(X_i,θ)
ShrinkageEstimator := RawEstimator → AdjustTowardTargetOrStructure → ReducedExpectedLoss
RegularisedEstimator := DataFitObjective + Penalty → ParameterEstimate
RidgeEstimator := argmin_β{||y-Xβ||²+λ||β||²_2}
LassoEstimator := argmin_β{||y-Xβ||²+λ||β||_1}

BayesianEstimation := PriorDistribution ∧ Likelihood → PosteriorDistribution → BayesEstimator
PriorDistribution := p(θ)
PosteriorDistribution := p(θ|x)∝p(x|θ)p(θ)
PosteriorMeanEstimator := E[θ|x] optimal_under squared-error loss
PosteriorMedianEstimator := Median(θ|x) optimal_under absolute-error loss under standard conditions
MAPEstimator := argmax_θp(θ|x)
BayesEstimator := argmin_a E[L(θ,a)|Data]
LossFunction := L(θ,a)
SquaredErrorLoss := (a-θ)^2
AbsoluteErrorLoss := |a-θ|
RiskFunction := R(θ,δ)=E_θ[L(θ,δ(X))]
MinimaxEstimator := Estimator minimising maximum risk over parameter space

ConfidenceInterval := [L(X),U(X)]
CoverageProbability := P_θ(L(X)≤θ≤U(X))
ConfidenceLevel := 1-α
WaldInterval := θ_hat±z_(1-α/2)SE(θ_hat)
TInterval := Estimate±t_(1-α/2,ν)EstimatedSE
ProfileLikelihood := TargetParameter → OptimiseOverNuisanceParameters → ProfileObjective

BootstrapEstimation := ObservedSample → ResampleWithReplacement → EstimatorReplicates → ApproximateSamplingDistribution
BootstrapBiasEstimate := Mean(θ_hat*)-θ_hat
BootstrapStandardError := SD(θ_hat*)
JackknifeEstimation := LeaveOneObservationOut → RecomputeEstimator → EstimateBiasAndVariance
DeltaMethod := AsymptoticDistributionOfEstimator → DistributionOfSmoothTransformation

NuisanceParameter := ModelParameter not primary target but required_for estimation
PlugInEstimator := FunctionalOfDistribution estimated_by substituting empirical_or_estimated distribution
PlugInPrinciple := θ=T(F) ⇒ θ_hat=T(F_n)
EstimatingEquation := Ψ_n(θ)=0
ZEstimator := SolutionToEstimatingEquation
Identifiability := DistinctParameterValues induce distinct observable distributions under model
NonIdentifiableModel := MultipleParameterValues imply same observable distribution
IdentificationConstraint := Estimation requires target parameter identifiable_from model_and_data
EstimatorRobustness := {InfluenceFunction,BreakdownPoint,SensitivityCurve}
EstimatorComparison := Compare(Bias,Variance,MSE,Consistency,Efficiency,Robustness,ComputationalCost)
BiasVarianceTradeoff := ReducedVariance may justify controlled bias when total expected loss decreases

MissingDataEstimation := MissingnessMechanism ∧ ObservedData ∧ EstimationMethod → ParameterEstimate
MultipleImputation := MultipleCompletedDatasets → SeparateEstimates → CombineEstimatesAndVariances
InverseProbabilityWeighting := WeightObservation by inverse estimated inclusion_or_response probability
SurveyEstimation := SamplingDesign ∧ Weights ∧ Stratification ∧ Clustering → DesignAwareEstimate
HorvitzThompsonEstimator := PopulationTotalEstimate=Σ_{i∈sample}y_i/π_i
FinitePopulationCorrection := SamplingWithoutReplacement ∧ FinitePopulation → VarianceAdjustment

PredictionEstimationDistinction := Estimation targets parameter_or_functional whereas Prediction targets unobserved outcome
EstimationUncertainty := SamplingVariation ∨ PosteriorUncertainty ∨ ModelUncertainty ∨ MeasurementUncertainty
ModelMisspecification := AssumedModel differs_from data-generating structure
SandwichVariance := RobustAsymptoticVariance under estimating-equation framework
RobustStandardError := sqrt(diagonal(SandwichVariance))

EstimatorConstraint := ConstraintOn(TargetParameter,Data,Model,SamplingDesign,ParameterSpace,LossFunction,RegularityConditions)
EstimatorValidity := TargetResolved ∧ ModelResolved ∧ DataCompatible ∧ IdentificationValid ∧ ComputationalProcedureValid
InvalidEstimation := UndefinedTarget ∨ NonIdentifiableParameter ∨ InvalidEstimatorDomain ∨ ViolatedCriticalAssumption ∨ IncorrectVarianceFormula ∨ UnrecordedSelectionOrWeighting ∨ FailedOptimisation

EstimationValidation :=
 ResolveTargetParameter → ResolvePopulationAndSample → ResolveStatisticalModel → ValidateIdentifiability
 → ResolveEstimator → ResolveParameterSpace → ValidateEstimatorDomain → ComputeEstimate
 → ResolveSamplingOrPosteriorDistribution → QuantifyUncertainty → EvaluateBiasVarianceAndMSE
 → DiagnoseRobustnessAndSensitivity → ValidateResult

EstimationState := {
 Population,Sample,Data,Model,Parameter,ParameterSpace,Estimand,Estimator,Estimate,
 Bias,Variance,StandardError,MSE,IntervalEstimate,Assumptions,Evidence,Confidence,Provenance
}

SEMANTIC_EQUIVALENCE(EstimationA,EstimationB) :=
 same(TargetParameterOrEstimand) ∧ same(DataSemantics) ∧ same(ModelSemantics)
 ∧ same(EstimatorDefinition) ∧ same(SamplingOrPosteriorFramework) ∧ same(UncertaintyConvention)

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

CoreClasses := {
 StatisticalEstimation,Parameter,Estimand,Estimator,Estimate,PointEstimator,IntervalEstimator,
 SamplingDistribution,Bias,Variance,StandardError,MeanSquaredError,MaximumLikelihoodEstimator,
 MethodOfMomentsEstimator,LeastSquaresEstimator,BayesianEstimator,RobustEstimator,
 BootstrapEstimator,StatisticalModel,Sample,Population
}

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

TypedRelations :=
 Estimator --estimates--> Parameter
 Estimator --targets--> Estimand
 Estimate --computedFrom--> Sample
 Estimator --assumesModel--> StatisticalModel
 Estimator --hasSamplingDistribution--> SamplingDistribution
 Estimator --hasBias--> Bias
 Estimator --hasVariance--> Variance
 Estimator --hasStandardError--> StandardError
 Estimator --hasMSE--> MeanSquaredError
 LeastSquaresEstimator --minimises--> SquaredErrorObjective
 MaximumLikelihoodEstimator --maximises--> Likelihood
 BayesianEstimator --conditionedOn--> PosteriorDistribution
 BootstrapEstimator --derivedFrom--> ResampledData
 Estimate --tracesTo--> Provenance

InferenceRules := {
 Estimator(T,θ) ∧ ObservedData(x) ⇒ Estimate=T(x),
 UnbiasedEstimator(T,θ) ⇒ E[T]=θ,
 Bias(T)=0 ⇒ MSE(T)=Var(T),
 ConsistentEstimator(T_n,θ) ⇒ T_n→θ under specified convergence,
 SamplingVariance(T) ⇒ StandardError(T)=sqrt(SamplingVariance(T)),
 LikelihoodAvailable ∧ ParameterIdentifiable ⇒ CandidateMLE=argmaxLikelihood,
 Prior ∧ Likelihood ⇒ Posterior,
 NonIdentifiableParameter ⇒ FlagEstimationInvalid
}

CanonicalEstimationRepresentation :=
 ParseEstimationQuestion → ResolvePopulation → ResolveSample → ResolveStatisticalModel
 → ResolveTargetParameterOrEstimand → ResolveParameterSpace → ValidateIdentifiability
 → ResolveEstimator → ComputeEstimate → ResolveSamplingOrPosteriorDistribution
 → QuantifyUncertainty → ValidateEstimatorProperties → PreserveProvenance

EstimationGraph := G=(V,E)

V := {
 Populations,Samples,Models,Parameters,Estimands,Estimators,Estimates,SamplingDistributions,
 Biases,Variances,StandardErrors,MeanSquaredErrors,ConfidenceIntervals,Priors,Posteriors,
 LossFunctions,Evidence,ProvenanceNodes
}

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

CONFIDENCE(EstimationResolution) :=
 weighted_mean(TargetResolution,PopulationSampleResolution,ModelResolution,IdentifiabilityValidity,
 EstimatorValidity,UncertaintyQuality,RobustnessAssessment,ProvenanceQuality)

AI_QUERY_INTERPRETATION :=
 Query → DetectStatisticalEstimationIntent → IdentifyPopulationSampleAndModel
 → IdentifyParameterOrEstimand → IdentifyPointIntervalFrequentistOrBayesianEstimation
 → ResolveRequestedEstimatorAndUncertainty

AI_REASONING_FLOW :=
 Query → ParseEstimationProblem → ResolvePopulation → ResolveSampleAndData → ResolveStatisticalModel
 → ResolveParameterOrEstimand → ResolveParameterSpace → ValidateIdentifiability → SelectEstimator
 → ComputeEstimate → ResolveSamplingDistributionOrPosterior → ComputeBiasVarianceMSEAndStandardErrorAsRequired
 → ConstructIntervalEstimateIfRequired → EvaluateRobustnessAndSensitivity → ValidateAssumptions
 → UpdateConfidence → PreserveProvenance → GenerateExplanation
AI_CONSUMPTION := {
 ParseStatisticalEstimationEntities,ParseTypedRelations,ResolvePopulations,ResolveSamples,
 ResolveModels,ResolveParameters,ResolveEstimands,ResolveParameterSpaces,ValidateIdentifiability,
 ResolveEstimators,ComputePointEstimates,ComputeMaximumLikelihoodEstimates,
 ComputeMethodOfMomentsEstimates,ComputeLeastSquaresEstimates,ComputeBayesianEstimates,
 ComputeRobustEstimates,ResolveSamplingDistributions,ComputeBias,ComputeVariance,
 ComputeStandardErrors,ComputeMeanSquaredErrors,ConstructConfidenceIntervals,
 ResolveBootstrapEstimation,ResolveJackknifeEstimation,ApplyDeltaMethod,
 EvaluateEstimatorEfficiency,EvaluateEstimatorRobustness,ValidateEstimation,
 CompareSemanticEquivalence,UpdateConfidence,PreserveProvenance,GenerateFormalExplanation
}

UPDATE_MODEL :=
 NewDataModelParameterEstimatorOrEvidence → ValidateInput → IdentifyAffectedEstimationClaims
 → UpdateEstimationState → RevalidateIdentifiability → RecomputeEstimate
 → RecomputeSamplingOrPosteriorDistribution → RecomputeBiasVarianceMSEAndSE
 → RecomputeIntervalEstimateIfRequired → ReevaluateRobustness → RecomputeConfidence
 → PreserveProvenance → VersionModel

DataUpdate :=
 ChangedSampleData
 ⇒ recompute(Estimate,StandardError,MSE,IntervalEstimate)

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

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

FLOW_HASH_INPUT :=
 canonical(
  ONTOLOGY,
  CoreClasses,
  CoreRelations,
  InferenceRules,
  StatisticalEstimation,
  EstimationValidation,
  EstimatorConstraint,
  SEMANTIC_EQUIVALENCE,
  AI_REASONING_FLOW
 )

FLOW_HASH := HASH(FLOW_HASH_INPUT)

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

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA