Bayesian Estimation Ontology

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

BEGIN_FLOW

BayesianEstimation :=
 PriorDistribution ∧ Likelihood ∧ PosteriorDistribution ∧ LossFunction
 ∧ BayesEstimator ∧ PosteriorUncertainty ∧ Validation

ObservedData := x={x_1,...,x_n}
Parameter := θ
ParameterVector := θ=(θ_1,...,θ_p)
ParameterSpace := Θ
StatisticalModel := p(x|θ)
PriorDistribution := π(θ)
Likelihood := L(θ|x)=p(x|θ)
JointDistribution := p(x,θ)=p(x|θ)π(θ)
MarginalLikelihood := p(x)=∫_Θp(x|θ)π(θ)dθ
Evidence := MarginalLikelihood

BayesTheorem :=
 p(θ|x)=p(x|θ)π(θ)/p(x)

PosteriorDistribution :=
 π(θ|x)∝L(θ|x)π(θ)

PosteriorNormalisation :=
 π(θ|x)=L(θ|x)π(θ)/∫_ΘL(u|x)π(u)du

PriorToPosteriorFlow :=
 PriorKnowledge ∧ ObservedData ∧ LikelihoodModel → PosteriorKnowledge

SequentialBayesianUpdating :=
 Posterior_t + NewData → Prior_(t+1) → Posterior_(t+1)

PriorPredictiveDistribution :=
 p(x)=∫p(x|θ)π(θ)dθ

PosteriorPredictiveDistribution :=
 p(x_new|x)=∫p(x_new|θ)π(θ|x)dθ

BayesEstimator :=
 δ_B(x)=argmin_a E[L(θ,a)|x]

LossFunction := L(θ,a)
PosteriorExpectedLoss := ρ(a|x)=E[L(θ,a)|x]
SquaredErrorLoss := (a-θ)^2
PosteriorMeanEstimator := E[θ|x] under squared-error loss
AbsoluteErrorLoss := |a-θ|
PosteriorMedianEstimator := Median(θ|x) under absolute-error loss
PosteriorModeEstimator := argmax_θπ(θ|x)
MAPEstimator := PosteriorModeEstimator

MLEVsMAP :=
 MLE=argmax_θL(θ|x)
 whereas
 MAP=argmax_θ[L(θ|x)π(θ)]

PosteriorMean := ∫θπ(θ|x)dθ
PosteriorVariance := ∫(θ-E[θ|x])²π(θ|x)dθ
PosteriorStandardDeviation := sqrt(PosteriorVariance)
PosteriorQuantile := Q_post(q)=inf{θ:F_post(θ)≥q}

CredibleInterval :=
 C such_that P(θ∈C|x)=1-α

EqualTailedCredibleInterval :=
 [Q_post(α/2),Q_post(1-α/2)]

HighestPosteriorDensityRegion :=
 {θ:π(θ|x)≥c}
 with posterior mass 1-α
 under applicable conditions

ConfidenceVsCredibleInterval :=
 ConfidenceInterval has repeated-sampling coverage interpretation
 whereas CredibleInterval has posterior-probability interpretation

PriorTypes := {
 InformativePrior,
 WeaklyInformativePrior,
 ReferencePrior,
 ConjugatePrior,
 HierarchicalPrior
}

PriorSensitivity :=
 ComparePosteriorResults across plausible prior specifications

PriorPredictiveCheck :=
 SampleFromPrior → SampleDataFromLikelihood → AssessPriorModelImplications

PosteriorPredictiveCheck :=
 SampleθFromPosterior → SampleReplicatedData → CompareReplicatedAndObservedData

BetaBinomialModel :=
 p~Beta(α,β) ∧ X|p~Binomial(n,p)

BetaBinomialPosterior :=
 p|X=x~Beta(α+x,β+n-x)

GammaPoissonModel :=
 λ~Gamma(α,β) ∧ X_i|λ~Poisson(λ)

GammaPoissonPosterior :=
 λ|x~Gamma(α+Σ_i x_i,β+n)
 under shape-rate parameterisation

GammaExponentialPosterior :=
 λ|x~Gamma(α+n,β+Σ_i x_i)

NormalNormalModel :=
 μ~N(μ0,τ0²)
 ∧ X_i|μ~N(μ,σ²)
 with known σ²

NormalPosteriorPrecision :=
 1/τ_n²=1/τ0²+n/σ²

NormalPosteriorMean :=
 μ_n=τ_n²[μ0/τ0²+nX_bar/σ²]

DirichletMultinomialPosterior :=
 p|x~Dirichlet(α_1+x_1,...,α_K+x_K)

HierarchicalBayesianModel :=
 DataLevel ∧ ParameterLevel ∧ HyperparameterLevel

Hyperparameter :=
 Parameter governing prior distribution

PartialPooling :=
 HierarchicalModel → GroupEstimatesShrinkTowardSharedStructure

EmpiricalBayes :=
 HyperparametersEstimatedFromData → PriorSpecification → PosteriorEstimation

BayesRisk :=
 r(π,δ)=∫R(θ,δ)π(θ)dθ

FrequentistRisk :=
 R(θ,δ)=E_θ[L(θ,δ(X))]

BayesRule :=
 DecisionRule minimising BayesRisk

PosteriorOdds :=
 PosteriorProbability(H1|x)/PosteriorProbability(H0|x)

PriorOdds :=
 PriorProbability(H1)/PriorProbability(H0)

BayesFactor :=
 p(x|H1)/p(x|H0)

PosteriorOddsRelation :=
 PosteriorOdds=BayesFactor×PriorOdds

ComputationalBayesianEstimation := {
 ClosedFormPosterior,
 NumericalIntegration,
 MonteCarlo,
 MarkovChainMonteCarlo,
 VariationalApproximation,
 LaplaceApproximation
}

MonteCarloEstimation :=
 θ^(1),...,θ^(M)~Posterior
 ⇒ E[g(θ)|x]≈(1/M)Σ_mg(θ^(m))

MarkovChainMonteCarlo :=
 ConstructMarkovChainWithTargetPosterior → GeneratePosteriorSamples

MetropolisHastings :=
 Proposal → AcceptanceProbability → MarkovChainTransition

GibbsSampling :=
 IterativelySampleFromFullConditionalDistributions

HamiltonianMonteCarlo :=
 GradientBasedDynamics → PosteriorExploration

MCMCConvergence :=
 Assess(ChainMixing,EffectiveSampleSize,Autocorrelation,BetweenChainAgreement)

MonteCarloStandardError :=
 NumericalUncertaintyOfPosteriorMonteCarloEstimate

VariationalInference :=
 ApproximationFamily q(θ)
 → Optimise KL(q(θ)||π(θ|x))

ELBO :=
 E_q[logp(x,θ)-logq(θ)]

LaplaceApproximation :=
 PosteriorMode ∧ LocalCurvature → GaussianApproximation

ImportanceSampling :=
 ProposalDistribution ∧ ImportanceWeights → PosteriorExpectationApproximation

SequentialMonteCarlo :=
 SequenceOfIntermediateDistributions → WeightedParticles → PosteriorApproximation

ApproximateBayesianComputation :=
 Simulator ∧ SummaryStatistics ∧ DistanceTolerance → ApproximatePosterior

PosteriorConsistency :=
 PosteriorMass concentrates_near true parameter_or_distribution
 as sample size increases under appropriate conditions

PriorDominance :=
 SmallSample ∨ WeakLikelihoodInformation
 ⇒ PriorMayStronglyInfluencePosterior

LikelihoodDominance :=
 HighlyInformativeData
 may dominate reasonable prior differences

ImproperPrior :=
 Prior not integrating_to one

ImproperPriorConstraint :=
 AllowedOnlyWhen resulting posterior is proper and inferential use justified

ProperPosterior :=
 ∫π(θ|x)dθ=1

ParameterisationSensitivity :=
 MAP and apparently flat priors
 may change under nonlinear reparameterisation

MissingDataBayesianEstimation :=
 MissingValuesAsLatentVariables
 → JointPosterior
 → MarginalPosteriorForTargets

DataAugmentation :=
 IntroduceLatentVariables → SimplifyConditionalSamplingOrInference

BayesianModelChecking :=
 PriorPredictiveChecks ∧ PosteriorPredictiveChecks ∧ SensitivityAnalysis

SimulationBasedCalibration :=
 DrawParameterFromPrior → SimulateData → FitPosterior → CheckRankUniformity

BayesianEstimationConstraint :=
 ConstraintOn(
  Parameter,
  ParameterSpace,
  Prior,
  Likelihood,
  PosteriorPropriety,
  LossFunction,
  ComputationalMethod
 )

BayesianEstimationConsistency :=
 Prior ∧ Likelihood ∧ ParameterSpace ∧ Posterior ∧ LossFunction ∧ Computation
 jointly_compatible

InvalidBayesianEstimation :=
 UndefinedPrior
 ∨ UndefinedLikelihood
 ∨ IncompatibleParameterSupport
 ∨ ImproperPosterior
 ∨ InvalidLossFunction
 ∨ FailedNumericalInference
 ∨ UnresolvedConvergenceFailure

BayesianEstimationValidation :=
 ResolveObservedData
 → ResolveParameterAndParameterSpace
 → ResolveLikelihood
 → ResolvePrior
 → ValidatePriorSupport
 → ComputePosteriorKernel
 → ValidatePosteriorPropriety
 → NormaliseOrApproximatePosterior
 → ResolveLossFunction
 → ComputeBayesEstimator
 → QuantifyPosteriorUncertainty
 → RunDiagnosticsAndSensitivityChecks
 → ValidateResult

BayesianEstimationState := {
 ObservedData,StatisticalModel,Parameter,ParameterSpace,PriorDistribution,
 Likelihood,MarginalLikelihood,PosteriorDistribution,LossFunction,BayesEstimator,
 PosteriorMean,PosteriorMedian,PosteriorMode,PosteriorVariance,CredibleInterval,
 PosteriorPredictiveDistribution,ComputationalMethod,DiagnosticStatus,
 Evidence,Confidence,Provenance
}

SEMANTIC_EQUIVALENCE(BayesianEstimateA,BayesianEstimateB) :=
 same(TargetParameterOrEstimand)
 ∧ same(DataSemantics)
 ∧ same(LikelihoodModel)
 ∧ same(PriorMeasure)
 ∧ same(LossFunction)
 ∧ equivalent(PosteriorDistribution)
 ∧ compatible(ComputationalApproximation)

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

CoreClasses := {
 BayesianEstimation,BayesEstimator,PriorDistribution,Likelihood,MarginalLikelihood,
 PosteriorDistribution,PosteriorPredictiveDistribution,PosteriorMean,PosteriorMedian,
 PosteriorMode,MAPEstimator,CredibleInterval,LossFunction,BayesRisk,BayesFactor,
 HierarchicalModel,Hyperparameter,MonteCarloEstimator,MarkovChainMonteCarlo,
 VariationalInference,Parameter,StatisticalModel
}

CoreRelations := {
 estimates,targets,computedFrom,hasPrior,hasLikelihood,hasPosterior,conditionedOn,
 minimises,predicts,hasHyperparameter,approximatedBy,updatedBy,constrainedBy,
 equivalentTo,tracesTo
}

TypedRelations :=
 BayesEstimator --estimates--> Parameter
 BayesEstimator --targets--> Estimand
 BayesianEstimation --hasPrior--> PriorDistribution
 BayesianEstimation --hasLikelihood--> Likelihood
 BayesianEstimation --hasPosterior--> PosteriorDistribution
 PosteriorDistribution --conditionedOn--> ObservedData
 BayesEstimator --minimises--> PosteriorExpectedLoss
 PosteriorDistribution --predicts--> PosteriorPredictiveDistribution
 HierarchicalModel --hasHyperparameter--> Hyperparameter
 PosteriorDistribution --approximatedBy--> ComputationalApproximation
 PriorDistribution --updatedBy--> Likelihood
 BayesEstimator --tracesTo--> Provenance

InferenceRules := {
 Prior ∧ Likelihood ∧ FiniteEvidence
  ⇒ Posterior∝Prior×Likelihood,

 SquaredErrorLoss
  ⇒ BayesEstimator=PosteriorMean,

 AbsoluteErrorLoss
  ⇒ BayesEstimator=PosteriorMedian,

 PosteriorDistribution ∧ FutureSamplingModel
  ⇒ PosteriorPredictiveDistribution,

 ImproperPosterior
  ⇒ FlagBayesianEstimationInvalid,

 MCMCMethod ∧ FailedConvergenceDiagnostics
  ⇒ FlagComputationalReliabilityRisk
}

CanonicalBayesianEstimationRepresentation :=
 ParseBayesianEstimationRequest
 → ResolveObservedData
 → ResolveTargetParameterOrEstimand
 → ResolveParameterSpace
 → ResolveStatisticalModelAndLikelihood
 → ResolvePriorDistribution
 → ValidatePriorSupport
 → ConstructPosteriorKernel
 → ValidatePosteriorPropriety
 → ResolveAnalyticOrComputationalInference
 → ComputePosteriorDistributionOrApproximation
 → ResolveLossFunction
 → ComputeBayesEstimator
 → ComputePosteriorUncertainty
 → ComputePosteriorPredictiveIfRequested
 → RunDiagnosticsAndSensitivityAnalysis
 → ValidateBayesianEstimation

BayesianEstimationGraph := G=(V,E)

V := {
 BayesianEstimations,BayesEstimators,Priors,Likelihoods,MarginalLikelihoods,
 PosteriorDistributions,PosteriorPredictiveDistributions,Parameters,ParameterSpaces,
 StatisticalModels,LossFunctions,PosteriorMeans,PosteriorMedians,PosteriorModes,
 CredibleIntervals,BayesFactors,HierarchicalModels,Hyperparameters,
 ComputationalApproximations,EvidenceNodes,ProvenanceNodes
}

E := {
 estimates,targets,computedFrom,hasPrior,hasLikelihood,hasPosterior,conditionedOn,
 minimises,predicts,hasHyperparameter,approximatedBy,updatedBy,constrainedBy,
 equivalentTo,tracesTo
}

CONFIDENCE(BayesianEstimationResolution) :=
 weighted_mean(
  TargetResolution,
  ModelLikelihoodResolution,
  PriorResolution,
  PosteriorProprietyValidity,
  LossFunctionResolution,
  ComputationalValidity,
  DiagnosticQuality,
  ProvenanceQuality
 )

AI_QUERY_INTERPRETATION :=
 Query
 → DetectBayesianEstimationIntent
 → IdentifyObservedDataAndTarget
 → ResolveLikelihoodAndPrior
 → IdentifyPosteriorPointIntervalPredictiveOrDecisionRequest
 → ResolveComputationalMethodAndDiagnostics

AI_REASONING_FLOW :=
 Query
 → ParseBayesianEstimationProblem
 → ResolveObservedData
 → ResolveTargetParameterOrEstimand
 → ResolveParameterSpace
 → ResolveLikelihood
 → ResolvePrior
 → ValidatePriorSupport
 → ConstructPosterior
 → ValidatePosteriorPropriety
 → SelectAnalyticOrComputationalMethod
 → ComputePosteriorApproximation
 → ResolveLossFunction
 → ComputeBayesEstimator
 → QuantifyPosteriorUncertainty
 → ComputeCredibleIntervalOrPosteriorPredictiveIfRequested
 → RunConvergencePredictiveAndSensitivityDiagnostics
 → ValidateResult
 → UpdateConfidence
 → PreserveProvenance
 → GenerateExplanation

AI_CONSUMPTION := {
 ParseBayesianEstimationEntities,ParseTypedRelations,ResolveObservedData,
 ResolveParameters,ResolveEstimands,ResolveParameterSpaces,ResolveStatisticalModels,
 ResolveLikelihoods,ResolvePriorDistributions,ComputeMarginalLikelihoods,
 ConstructPosteriorDistributions,ValidatePosteriorPropriety,ComputePosteriorMeans,
 ComputePosteriorMedians,ComputePosteriorModes,ComputeMAPEstimates,
 ComputePosteriorVariances,ComputeCredibleIntervals,ComputePosteriorPredictiveDistributions,
 ResolveConjugateModels,ResolveHierarchicalModels,ComputeBayesFactors,
 ExecuteMonteCarloEstimation,ExecuteMCMC,ExecuteVariationalInference,
 ExecuteLaplaceApproximation,ExecuteImportanceSampling,EvaluateMCMCDiagnostics,
 PerformPriorSensitivityAnalysis,PerformPriorPredictiveChecks,
 PerformPosteriorPredictiveChecks,ValidateBayesianEstimation,
 CompareSemanticEquivalence,UpdateConfidence,PreserveProvenance,
 GenerateFormalExplanation
}

UPDATE_MODEL :=
 NewDataPriorLikelihoodLossOrEvidence
 → ValidateInput
 → IdentifyAffectedBayesianClaims
 → UpdateBayesianEstimationState
 → RecomputePosteriorKernel
 → RevalidatePosteriorPropriety
 → RecomputePosteriorOrApproximation
 → RecomputeBayesEstimator
 → RecomputePosteriorUncertainty
 → RecomputePosteriorPredictiveIfRequired
 → RerunDiagnosticsAndSensitivity
 → RecomputeConfidence
 → PreserveProvenance
 → VersionModel

DataUpdate :=
 ChangedObservedData
 ⇒ recompute(
  Likelihood,
  MarginalLikelihood,
  PosteriorDistribution,
  BayesEstimator,
  CredibleInterval,
  PosteriorPredictiveDistribution
 )

PriorUpdate :=
 ChangedPriorDistribution
 ⇒ recompute(
  PosteriorDistribution,
  MarginalLikelihood,
  BayesEstimator,
  PriorSensitivityStatus
 )

LikelihoodUpdate :=
 ChangedLikelihoodModel
 ⇒ reevaluate(
  PosteriorDistribution,
  ModelFit,
  PredictiveDistribution,
  ComputationalMethod
 )

FLOW_HASH_INPUT :=
 canonical(
  ONTOLOGY,
  CoreClasses,
  CoreRelations,
  InferenceRules,
  BayesianEstimation,
  BayesianEstimationValidation,
  BayesianEstimationConstraint,
  SEMANTIC_EQUIVALENCE,
  AI_REASONING_FLOW
 )

FLOW_HASH := HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {
 BayesianEstimationEntityResolution,
 ObservedData,
 StatisticalModel,
 TargetParameter,
 Estimand,
 ParameterSpace,
 PriorDistribution,
 Likelihood,
 MarginalLikelihood,
 PosteriorDistribution,
 LossFunction,
 BayesEstimator,
 PosteriorMean,
 PosteriorMedian,
 PosteriorMode,
 PosteriorVariance,
 CredibleInterval,
 PosteriorPredictiveDistribution,
 ComputationalMethod,
 ConvergenceStatus,
 PriorSensitivityStatus,
 PosteriorPredictiveStatus,
 ValidationStatus,
 Confidence,
 Provenance,
 Explanation
}

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA