Bayesian Statistics Ontology

FULL EXECUTABLE FLOW ALGEBRA — BAYESIAN STATISTICS ONTOLOGY
Kireth Research — Semantic Library
URL: https://kirethresearch.org/booksai-core/ontologies/probability-statistics-ontologies/bayesian-statistics/bayesian-statistics-ontology-2.html
FLOW_VERSION := "4.0"
FLOW_TYPE := "Full Executable Flow Algebra"
DOMAIN := "Probability & Statistics Ontologies"
SUBDOMAIN := "Bayesian Statistics"
ONTOLOGY := "Bayesian Statistics Ontology"
PRIMARY_ENTITY := BayesianStatistics
AUTHORSHIP := "Mark Higgins — Sky (Recursive AI)"
SOURCE_SITE := "https://kirethresearch.org"

BEGIN_FLOW

BayesianStatistics := StatisticalInference∧ProbabilityModel∧PriorInformation∧LikelihoodEvidence∧PosteriorUpdating∧PredictiveReasoning
Parameter := θ
ParameterSpace := Θ
ObservedData := D
FutureData := D_new
StatisticalModel := M
Prior := p(θ|M)
Likelihood := p(D|θ,M)
Posterior := p(θ|D,M)
Evidence := p(D|M)
BayesRule := p(θ|D,M)=p(D|θ,M)p(θ|M)/p(D|M)
MarginalLikelihood := p(D|M)=∫p(D|θ,M)p(θ|M)dθ
PosteriorKernel := p(θ|D,M)∝p(D|θ,M)p(θ|M)
BayesianUpdate := Prior×Likelihood→Posterior
SequentialBayesianUpdate := PosteriorAfterD1→PriorForD2→PosteriorAfterD1AndD2
PriorDistribution := ProbabilityDistributionRepresentingPreDataUncertainty
InformativePrior := PriorEncodingSubstantiveInformation
WeaklyInformativePrior := PriorProvidingRegularisationWithoutStrongConcentration
ImproperPrior := PriorWithoutFiniteNormalisingConstant
PosteriorProprietyConstraint := ImproperPriorAllowedOnlyIfPosteriorProper
PriorPredictive := p(D)=∫p(D|θ)p(θ)dθ
PriorPredictiveCheck := DrawθFromPrior→DrawReplicatedData→AssessPlausibility
LikelihoodFunction := θ→p(D|θ)
LogLikelihood := ℓ(θ)=log p(D|θ)
ConditionalIndependence := p(D|θ)=Π_i p(y_i|θ) when applicable
PosteriorMean := E[θ|D]
PosteriorMedian := Q_0.5(θ|D)
PosteriorMode := argmax_θp(θ|D)
MAP := argmax_θp(D|θ)p(θ)
PosteriorVariance := Var(θ|D)
PosteriorStandardDeviation := sqrt(Var(θ|D))
CredibleInterval := C suchThat P(θ∈C|D)=1-α
EqualTailedInterval := [Q_(α/2),Q_(1-α/2)]
HighestDensityInterval := RegionContainingSpecifiedPosteriorMassWithHighestDensity
PriorOdds := P(H1)/P(H0)
PosteriorOdds := P(H1|D)/P(H0|D)
BayesFactor := BF_10=p(D|H1)/p(D|H0)
PosteriorOddsIdentity := PosteriorOdds=PriorOdds×BayesFactor
ModelPrior := p(M)
PosteriorModelProbability := p(M|D)∝p(D|M)p(M)
PosteriorPredictive := p(D_new|D)=∫p(D_new|θ)p(θ|D)dθ
PosteriorPredictiveMean := E[D_new|D]
PosteriorPredictiveVariance := Var(D_new|D)
PosteriorPredictiveCheck := D_rep~p(D_rep|D)→CompareWithObservedData
Conjugacy := PriorAndLikelihoodYieldPosteriorInSameParametricFamily
BetaBinomialConjugacy := Beta(α,β)∧Binomial(n,p)→Beta(α+x,β+n-x)
GammaPoissonConjugacy := GammaPrior∧PoissonLikelihood→GammaPosterior
NormalNormalConjugacy := NormalPrior∧NormalLikelihood→NormalPosterior
HierarchicalBayesianModel := Hyperprior→Hyperparameter→GroupParameter→Observation
Hyperparameter := η
Hyperprior := p(η)
PartialPooling := GroupSpecificPosteriorEstimatesBorrowInformationAcrossGroups
CompletePooling := SharedParameterAcrossGroups
NoPooling := IndependentParametersAcrossGroups
Shrinkage := PosteriorGroupEstimateMovesTowardPopulationStructure
Exchangeability := JointPriorOrSamplingStructureInvariantToGroupPermutationUnderModel
BayesianLinearRegression := p(β,σ²|D)∝p(D|β,σ²)p(β,σ²)
BayesianLogisticRegression := BernoulliLikelihood∧LogitLink∧PriorOnCoefficients
BayesianTimeSeries := DynamicLikelihood∧PriorOnStatesAndParameters
LatentVariable := UnobservedQuantityInGenerativeModel
LatentPosterior := p(Z,θ|D)
AnalyticInference := ClosedFormPosteriorOrIntegralsAvailable
GridApproximation := EvaluatePosteriorOnGrid→Normalise→Summarise
LaplaceApproximation := GaussianApproximationAroundPosteriorMode
VariationalInference := OptimiseApproximateDistribution→ApproximatePosterior
ImportanceSampling := ProposalDraws→ImportanceWeights→WeightedPosteriorApproximation
MonteCarloExpectation := E[f(θ)|D]≈S^(-1)Σ_sf(θ_s)
MCMC := ConstructMarkovChainWithPosteriorInvariantDistribution
MetropolisHastings := Propose→ComputeAcceptanceProbability→AcceptOrReject→Iterate
GibbsSampling := SequentiallySampleFullConditionalDistributions
HamiltonianMonteCarlo := HamiltonianDynamicsUsingPosteriorGradient→Proposal
NUTS := AdaptiveTrajectoryLengthHamiltonianMonteCarlo
EffectiveSampleSize := InformationEquivalentToIndependentPosteriorDraws
RHat := PotentialScaleReductionDiagnosticAcrossChains
MonteCarloStandardError := NumericalSamplingUncertaintyInPosteriorSummary
Divergence := HMCDiagnosticForProblematicNumericalTrajectory
ConvergenceAssessment := RHat∧ESS∧TracePlots∧Divergences
ConvergenceConstraint := FailedDiagnostics→PosteriorSummariesNotYetReliable
Identifiability := ParametersDistinguishableThroughLikelihoodAndModel
WeakIdentification := DataProvideLimitedInformationAboutParameter
Nonidentifiability := DistinctParameterConfigurationsYieldEquivalentObservationalDistribution
PriorRegularisation := PriorConstrainsWeaklyIdentifiedDirections
PriorSensitivity := PosteriorDependenceOnReasonableAlternativePriors
PriorSensitivityAnalysis := FitAlternativePriors→ComparePosteriorQuantities
LikelihoodSensitivity := FitAlternativeDataModels→ComparePosteriorQuantities
ModelSensitivity := CompareReasonableStructuralAssumptions
RobustBayesianAnalysis := HeavyTailedLikelihoods∨Mixtures∨SensitivityAnalysis
DecisionTheory := Posterior∧ActionSpace∧LossFunction
PosteriorExpectedLoss := E[L(a,θ)|D]
BayesAction := argmin_aE[L(a,θ)|D]
SquaredErrorBayesAction := PosteriorMean
AbsoluteErrorBayesAction := PosteriorMedian
PosteriorExpectedUtility := E[U(a,θ)|D]
BayesianModelComparison := CandidateModels→PredictiveOrMarginalEvidence→Comparison
BayesFactorSensitivity := MarginalLikelihoodMayBeStronglyAffectedByPriorScale
LOOCV := LeaveOneOutCrossValidation
PSISLOO := ParetoSmoothedImportanceSamplingLOOApproximation
WAIC := WidelyApplicableInformationCriterion
BayesianModelAveraging := PosteriorModelWeights→MixturePredictiveDistribution
Stacking := PredictiveWeightsChosenToOptimiseOutOfSamplePerformance
Calibration := PredictiveProbabilitiesMatchObservedFrequenciesAcrossRepeatedUse
Sharpness := PredictiveConcentrationSubjectToCalibration
LogScore := -log p(y|predictive)
CRPS := ContinuousRankedProbabilityScore
ROPE := RegionOfPracticalEquivalence
PosteriorSignProbability := P(θ>0|D)OrP(θ<0|D)
BayesianMissingDataModel := MissingValues∧ObservedValues∧JointModel→PosteriorImputation
MeasurementErrorModel := LatentTrueValue∧ObservationErrorDistribution
MixtureModel := Σ_kπ_kp(D|θ_k)
CredibleVsConfidenceInterval := CredibleIntervalIsPosteriorProbabilityRegionWhereasConfidenceIntervalHasRepeatedSamplingCoverageInterpretation
PValueConstraint := PValue≠P(NullHypothesis|Data)
CausalConstraint := PosteriorAssociationAlone↛CausalEffect
PosteriorDoesNotGuaranteeTruth := ValidityDependsOnModelPriorDataAndComputation

BayesianConstraint := ConstraintOn(ParameterSpace,Prior,Likelihood,PosteriorPropriety,Identifiability,Computation,DecisionTarget,PredictiveTarget)
BayesianConsistency := Parameterisation∧Prior∧Likelihood∧Data∧Computation jointly_compatible
InvalidBayesianAnalysis := UndefinedPrior∨InvalidLikelihood∨ImproperPosterior∨SevereNonidentifiabilityUnresolved∨ComputationalNonconvergence∨PosteriorPredictiveFailureUnaddressed∨UnsupportedCausalInterpretation
BayesianValidation := ResolveScientificQuestion→DefineGenerativeModel→ResolveParametersAndParameterSpace→SpecifyLikelihood→SpecifyPrior→RunPriorPredictiveChecks→ObserveData→ComputePosterior→DiagnoseComputation→SummarisePosterior→GeneratePosteriorPredictiveDistribution→RunPosteriorPredictiveChecks→AssessPriorLikelihoodAndModelSensitivity→EvaluatePredictivePerformance→ResolveDecisionOrScientificInterpretation→ValidateResult
BayesianState := {StatisticalModel,Parameters,ParameterSpace,Hyperparameters,Prior,Likelihood,Posterior,Evidence,PosteriorMean,PosteriorMedian,PosteriorMode,PosteriorVariance,CredibleIntervals,PosteriorPredictive,BayesFactors,MCMCChains,RHat,EffectiveSampleSize,MonteCarloStandardError,Divergences,PriorPredictiveStatus,PosteriorPredictiveStatus,SensitivityStatus,PredictivePerformance,DecisionTarget,EvidenceState,Confidence,Provenance}
SEMANTIC_EQUIVALENCE(BayesianModelA,BayesianModelB) := same(ParameterSemantics)∧same(DataSemantics)∧same(LikelihoodOrEquivalentGenerativeDistribution)∧same(PriorInformationContent)∧same(PosteriorTarget)∧same(PredictiveOrDecisionTarget)

BayesianStatisticsOntology := Classes∧Relations∧Constraints∧Axioms∧InferenceRules
CoreClasses := {BayesianStatistics,StatisticalModel,Parameter,ParameterSpace,PriorDistribution,Likelihood,PosteriorDistribution,MarginalLikelihood,PosteriorMean,CredibleInterval,PosteriorPredictiveDistribution,BayesFactor,ConjugatePrior,HierarchicalBayesianModel,Hyperparameter,LatentVariable,MCMC,MetropolisHastings,GibbsSampling,HamiltonianMonteCarlo,NUTS,EffectiveSampleSize,RHat,BayesianDecision,PosteriorPredictiveCheck}
CoreRelations := {hasParameter,hasPrior,hasLikelihood,updatesTo,normalisedBy,hasPosterior,hasEvidence,predicts,hasCredibleInterval,comparesModel,hasHyperparameter,hasLatentVariable,sampledBy,diagnosedBy,evaluatedBy,optimises,conditionedOn,constrainedBy,equivalentTo,tracesTo}
TypedRelations := BayesianModel--hasParameter-->Parameter ∧ Parameter--hasPrior-->PriorDistribution ∧ BayesianModel--hasLikelihood-->Likelihood ∧ PriorAndLikelihood--updatesTo-->PosteriorDistribution ∧ PosteriorDistribution--normalisedBy-->MarginalLikelihood ∧ BayesianModel--hasPosterior-->PosteriorDistribution ∧ PosteriorDistribution--predicts-->FutureData ∧ Parameter--hasCredibleInterval-->CredibleInterval ∧ BayesianModel--comparesModel-->BayesFactor ∧ PosteriorDistribution--sampledBy-->MCMC ∧ MCMC--diagnosedBy-->ConvergenceDiagnostic ∧ BayesianModel--evaluatedBy-->PosteriorPredictiveCheck ∧ BayesianAnalysis--tracesTo-->Provenance
InferenceRules := {Prior∧Likelihood⇒Posterior,ConjugatePrior∧CompatibleLikelihood⇒ClosedFormPosteriorFamily,SquaredErrorLoss⇒PosteriorMeanBayesAction,AbsoluteErrorLoss⇒PosteriorMedianBayesAction,PosteriorPredictive⇒IntegrateFutureLikelihoodOverPosterior,RHatHigh∨ESSLow∨DivergencesPresent⇒FlagComputationalReliabilityRisk,PriorSensitivityHigh⇒FlagPriorDependence,PosteriorPredictiveFailure⇒FlagModelAdequacyRisk,BayesianAssociation↛CausalEffect}
CanonicalBayesianStatisticsRepresentation := ParseBayesianRequest→ResolveModelParametersAndData→SpecifyLikelihood→SpecifyPrior→ExecutePriorPredictiveChecks→ComputePosterior→SelectAnalyticApproximateOrSamplingInference→DiagnoseComputation→ComputePosteriorSummariesAndCredibleIntervals→GeneratePosteriorPredictiveDistribution→ExecutePosteriorPredictiveChecks→AssessPriorLikelihoodAndModelSensitivity→EvaluatePredictivePerformance→ResolveDecisionTarget→ValidateBayesianStatistics
BayesianStatisticsGraph := G=(V,E)
V := {BayesianStatisticsNodes,StatisticalModels,Parameters,ParameterSpaces,PriorDistributions,Likelihoods,PosteriorDistributions,MarginalLikelihoods,PosteriorMeans,CredibleIntervals,PosteriorPredictiveDistributions,BayesFactors,ConjugatePriors,HierarchicalBayesianModels,Hyperparameters,LatentVariables,MCMCChains,MetropolisHastingsSamplers,GibbsSamplers,HamiltonianMonteCarloSamplers,NUTSSamplers,EffectiveSampleSizes,RHatDiagnostics,BayesianDecisions,PosteriorPredictiveChecks,EvidenceNodes,ProvenanceNodes}
E := {hasParameter,hasPrior,hasLikelihood,updatesTo,normalisedBy,hasPosterior,hasEvidence,predicts,hasCredibleInterval,comparesModel,hasHyperparameter,hasLatentVariable,sampledBy,diagnosedBy,evaluatedBy,optimises,conditionedOn,constrainedBy,equivalentTo,tracesTo}
CONFIDENCE(BayesianStatisticsResolution) := weighted_mean(ModelResolution,ParameterResolution,PriorValidity,LikelihoodValidity,PosteriorPropriety,IdentifiabilityValidity,ComputationalValidity,ConvergenceQuality,PriorPredictiveQuality,PosteriorPredictiveQuality,SensitivityAssessment,PredictiveValidation,ProvenanceQuality)
AI_QUERY_INTERPRETATION := Query→DetectBayesianStatisticsIntent→IdentifyModelParametersDataAndPriorContext→ResolveEstimationPredictionHypothesisDecisionOrModelComparisonIntent→IdentifyRequestedPosteriorCredibleIntervalBayesFactorPredictiveOrComputationalQuantity→ResolveSensitivityAndValidationRequirements
AI_REASONING_FLOW := Query→ParseBayesianStatisticsProblem→ResolveModelParametersAndData→SpecifyLikelihood→SpecifyPrior→ExecutePriorPredictiveReasoning→ComputePosterior→SelectInferenceMethod→DiagnoseComputation→ComputePosteriorSummaries→ComputeCredibleIntervals→GeneratePosteriorPredictiveDistribution→ExecutePosteriorPredictiveChecks→AssessPriorLikelihoodAndModelSensitivity→EvaluatePredictivePerformance→ResolveDecisionOrScientificInterpretation→UpdateConfidence→PreserveProvenance→GenerateExplanation
AI_CONSUMPTION := {ParseBayesianStatisticsEntities,ParseTypedRelations,ResolveStatisticalModels,ResolveParameters,ResolveParameterSpaces,ResolveLikelihoods,ResolvePriors,ComputeBayesRule,ComputeMarginalLikelihoods,ComputePosteriorDistributions,ComputePosteriorMeans,ComputePosteriorMedians,ComputePosteriorModes,ComputePosteriorVariances,ComputeCredibleIntervals,ComputeHighestDensityIntervals,ComputeBayesFactors,GeneratePriorPredictiveSamples,GeneratePosteriorPredictiveSamples,FitConjugateModels,FitHierarchicalBayesianModels,ResolveLatentVariables,ExecuteGridApproximation,ExecuteLaplaceApproximation,ExecuteVariationalInference,ExecuteImportanceSampling,ExecuteMetropolisHastings,ExecuteGibbsSampling,ExecuteHamiltonianMonteCarlo,ExecuteNUTS,ComputeEffectiveSampleSize,ComputeRHat,ComputeMonteCarloStandardError,DetectDivergences,ExecutePriorSensitivityAnalysis,ExecuteLikelihoodSensitivityAnalysis,ExecutePosteriorPredictiveChecks,ExecutePSISLOO,ComputeWAIC,ResolveBayesianDecisionRules,ValidateBayesianStatisticsSemantics,CompareSemanticEquivalence,UpdateConfidence,PreserveProvenance,GenerateFormalExplanation}
UPDATE_MODEL := NewDataPriorLikelihoodModelOrEvidence→ValidateInput→IdentifyAffectedBayesianClaims→UpdateBayesianState→RecomputePriorPredictiveIfRequired→RecomputePosterior→RerunComputationalDiagnostics→RecomputePosteriorSummariesAndPredictiveDistribution→RerunSensitivityAndPredictiveChecks→RecomputeConfidence→PreserveProvenance→VersionModel
DataUpdate := NewObservedData⇒recompute(Likelihood,Posterior,PosteriorSummaries,PosteriorPredictive)
PriorUpdate := ChangedPrior⇒recompute(PriorPredictive,Posterior,CredibleIntervals,SensitivityStatus)
LikelihoodUpdate := ChangedLikelihood⇒reevaluate(Posterior,ModelFit,PosteriorPredictiveChecks,PredictivePerformance)
ComputationUpdate := ChangedInferenceMethod⇒reevaluate(MonteCarloError,ConvergenceDiagnostics,PosteriorApproximationQuality)
FLOW_HASH_INPUT := canonical(ONTOLOGY,CoreClasses,CoreRelations,InferenceRules,BayesianStatistics,BayesianValidation,BayesianConstraint,SEMANTIC_EQUIVALENCE,AI_REASONING_FLOW)
FLOW_HASH := HASH(FLOW_HASH_INPUT)
OUTPUT_MODEL := {BayesianStatisticsEntityResolution,StatisticalModel,Parameters,ParameterSpace,Hyperparameters,Prior,Likelihood,Posterior,Evidence,PosteriorMean,PosteriorMedian,PosteriorMode,PosteriorVariance,CredibleIntervals,PosteriorPredictive,BayesFactors,MCMCChains,RHat,EffectiveSampleSize,MonteCarloStandardError,Divergences,PriorPredictiveStatus,PosteriorPredictiveStatus,SensitivityStatus,PredictivePerformance,DecisionTarget,ValidationStatus,Confidence,Provenance,Explanation}

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA