Bayesian Inference Ontology

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

BEGIN_FLOW

BayesianInference := PriorUncertainty∧LikelihoodEvidence∧PosteriorUpdating∧PredictiveInference∧DecisionReasoning
Parameter := θ
ParameterSpace := Θ
ObservedData := D
FutureData := D_new
Model := 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)=∫_Θp(D|θ,M)p(θ|M)dθ
PosteriorKernel := p(θ|D,M)∝p(D|θ,M)p(θ|M)
InferenceUpdate := Prior×Likelihood→Posterior
SequentialInference := Posterior(D_1)→PriorForD_2→Posterior(D_1,D_2)

PriorInformation := InformationEncodedBeforeCurrentData
InformativePrior := SubstantivelyConcentratedPrior
WeaklyInformativePrior := RegularisingPriorWithBroadPlausibleSupport
DiffusePrior := BroadPrior
ImproperPrior := NonNormalisablePrior
PosteriorPropriety := ∫p(D|θ)p(θ)dθ<∞
PriorPredictive := p(D)=∫p(D|θ)p(θ)dθ
PriorPredictiveCheck := SimulatePriorPredictive→AssessPlausibility

LikelihoodFunction := θ→p(D|θ)
LogLikelihood := ℓ(θ)=logp(D|θ)
ConditionalIndependence := p(D|θ)=Π_ip(y_i|θ) when model permits
PosteriorDistribution := UpdatedProbabilityDistributionOverParameterSpace
PosteriorMean := E[θ|D]
PosteriorMedian := Median(θ|D)
PosteriorMode := argmax_θp(θ|D)
MAPEstimate := argmax_θp(D|θ)p(θ)
PosteriorVariance := Var(θ|D)
PosteriorQuantile := Q_q(θ|D)
CredibleInterval := C:P(θ∈C|D)=1-α
EqualTailedInterval := [Q_(α/2),Q_(1-α/2)]
HighestDensityInterval := HighestDensityRegionContainingSpecifiedPosteriorMass

PosteriorProbability := P(H|D)
PriorOdds := P(H1)/P(H0)
PosteriorOdds := P(H1|D)/P(H0|D)
BayesFactor := BF_10=p(D|H1)/p(D|H0)
PosteriorOddsRelation := PosteriorOdds=PriorOdds×BayesFactor
ModelPosterior := 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)
PosteriorPredictiveInterval := ProbabilityRegionForFutureObservation
PosteriorPredictiveCheck := D_rep~p(D_rep|D)→Compare(D_rep,D)

ConjugateInference := ConjugatePrior∧CompatibleLikelihood→ClosedFormPosterior
BetaBinomialInference := Beta(α,β)∧Binomial(n,p)→Beta(α+x,β+n-x)
GammaPoissonInference := GammaPrior∧PoissonLikelihood→GammaPosterior
NormalNormalInference := NormalPrior∧NormalLikelihood→NormalPosteriorUnderCompatibleVarianceSpecification

HierarchicalInference := Hyperprior→Hyperparameter→GroupParameters→Observations→JointPosterior
Hyperparameter := η
Hyperprior := p(η)
GroupParameter := θ_j
PartialPooling := HierarchicalPosteriorSharesInformationAcrossGroups
Shrinkage := GroupPosteriorMovesTowardPopulationStructure
Exchangeability := GroupLabelsPermutableUnderSpecifiedModel

LatentVariable := Z
JointPosterior := p(θ,Z|D)
LatentInference := InferUnobservedZAndθFromObservedData
MissingDataInference := MissingValuesAsLatentVariables→PosteriorImputation
MeasurementErrorInference := LatentTrueQuantity∧ObservationModel→PosteriorForTrueQuantity

AnalyticInference := PosteriorAndIntegralsAvailableInClosedForm
ApproximateInference := PosteriorRequiresNumericalApproximation
GridInference := EvaluatePosteriorOnGrid→Normalise→Summarise
LaplaceInference := GaussianApproximationAroundPosteriorMode
VariationalInference := Optimiseq(θ)ToApproximatePosterior
ImportanceSampling := ProposalSamples→Weights→PosteriorApproximation
MonteCarloInference := PosteriorSamples→ApproximateExpectations
MCMC := MarkovChainWithPosteriorInvariantDistribution
MetropolisHastings := Proposal→AcceptanceProbability→AcceptOrReject
GibbsSampling := IterateFullConditionalDraws
HamiltonianMonteCarlo := GradientDrivenHamiltonianProposal
NUTS := AdaptiveHamiltonianMonteCarloTrajectorySelection

MCMCChain := {θ^(1),...,θ^(S)}
Warmup := AdaptationAndInitialisationIterations
ChainAutocorrelation := DependenceAcrossMCMCDraws
EffectiveSampleSize := IndependentEquivalentInformationInCorrelatedDraws
RHat := BetweenWithinChainConvergenceDiagnostic
MonteCarloStandardError := NumericalUncertaintyOfPosteriorEstimate
Divergence := HMCTrajectoryDiagnostic
ConvergenceAssessment := RHat∧ESS∧TracePlots∧Divergences
ComputationalValidity := ConvergenceDiagnosticsWithinAcceptableCriteria

PosteriorInferenceTarget := Parameter∨FunctionOfParameter∨Prediction∨Model∨Decision
DerivedQuantity := g(θ)
DerivedPosterior := DistributionOfg(θ)UnderPosterior
PosteriorTransformation := θ^(s)→g(θ^(s))→PosteriorSummaryOfDerivedQuantity

DecisionTheory := Posterior∧ActionSpace∧LossFunction
PosteriorExpectedLoss := E[L(a,θ)|D]
BayesAction := argmin_aE[L(a,θ)|D]
SquaredErrorLoss⇒PosteriorMeanOptimal
AbsoluteErrorLoss⇒PosteriorMedianOptimal
ExpectedUtility := E[U(a,θ)|D]

HypothesisInference := PosteriorModelProbability∨BayesFactor∨PosteriorRegionProbability
ROPE := RegionOfPracticalEquivalence
ROPEProbability := P(θ∈ROPE|D)
SignProbability := P(θ>0|D)OrP(θ<0|D)
PracticalEffectThreshold := DomainDefinedEffectMagnitude

PriorSensitivity := PosteriorVariationAcrossReasonablePriors
PriorSensitivityAnalysis := RefitAlternativePriors→ComparePosterior
LikelihoodSensitivity := RefitAlternativeLikelihoods→ComparePosterior
ModelSensitivity := RefitAlternativeModelStructures→ComparePosterior
RobustInference := ConclusionsStableAcrossReasonableModellingChoices

Identifiability := ParametersDistinguishableByObservationalDistribution
WeakIdentification := LikelihoodProvidesLimitedParameterInformation
Nonidentifiability := DistinctParametersYieldEquivalentObservationalDistribution
PriorRegularisation := PriorStabilisesWeaklyIdentifiedDirections

PosteriorPredictiveAdequacy := ReplicatedDataReproduceRelevantObservedFeatures
ModelMisspecification := ModelFailsToRepresentImportantGeneratingStructure
PosteriorPredictiveFailure := SystematicMismatchBetweenReplicatedAndObservedData

PredictiveEvaluation := PosteriorPredictiveDistribution→OutOfSampleScoring
LOO := LeaveOneOutPredictiveEvaluation
PSISLOO := ParetoSmoothedImportanceSamplingApproximationToLOO
WAIC := WidelyApplicableInformationCriterion
LogScore := -logp(y|predictive)
CRPS := ContinuousRankedProbabilityScore

BayesianInferenceVsFrequentistInference := PosteriorConditionalProbabilityReasoningVsRepeatedSamplingInference
CredibleIntervalVsConfidenceInterval := PosteriorProbabilityRegionVsRepeatedSamplingCoverageProcedure
PValueConstraint := PValue≠PosteriorProbabilityOfNull
BayesianInferenceDoesNotGuaranteeTruth := PosteriorConditionalOnModelPriorAndData
CausalInferenceConstraint := BayesianPosteriorAssociation↛CausalEffectWithoutIdentificationAssumptions

BayesianInferenceConstraint := ConstraintOn(ParameterSpace,Prior,Likelihood,PosteriorPropriety,Identifiability,Computation,PredictiveTarget,DecisionTarget)
BayesianInferenceConsistency := Parameterisation∧Prior∧Likelihood∧ObservedData∧InferenceMethod jointly_compatible
InvalidBayesianInference := UndefinedPrior∨InvalidLikelihood∨ImproperPosterior∨UnresolvedNonidentifiability∨ComputationalNonconvergence∨SeverePosteriorPredictiveFailureUnaddressed∨UnsupportedCausalInterpretation

BayesianInferenceValidation := ResolveScientificQuestion→DefineModelAndParameters→SpecifyLikelihood→SpecifyPrior→ExecutePriorPredictiveChecks→ObserveData→ComputePosterior→DiagnoseComputation→ComputePosteriorSummaries→GenerateCredibleIntervals→GeneratePosteriorPredictiveDistribution→ExecutePosteriorPredictiveChecks→AssessPriorLikelihoodAndModelSensitivity→EvaluatePredictivePerformance→ResolveDecisionOrScientificInterpretation→ValidateResult

BayesianInferenceState := {Model,Parameters,ParameterSpace,Prior,Likelihood,Posterior,Evidence,PosteriorMean,PosteriorMedian,PosteriorMode,PosteriorVariance,CredibleIntervals,PosteriorPredictive,BayesFactors,LatentVariables,MCMCChains,RHat,EffectiveSampleSize,MonteCarloStandardError,Divergences,PriorPredictiveStatus,PosteriorPredictiveStatus,SensitivityStatus,PredictivePerformance,DecisionTarget,Confidence,Provenance}

SEMANTIC_EQUIVALENCE(InferenceA,InferenceB) := same(ParameterSemantics)∧same(DataSemantics)∧same(LikelihoodOrEquivalentGenerativeModel)∧same(PriorInformationContent)∧same(PosteriorTarget)∧same(PredictiveOrDecisionTarget)

BayesianInferenceOntology := Classes∧Relations∧Constraints∧Axioms∧InferenceRules
CoreClasses := {BayesianInference,Parameter,Prior,Likelihood,Posterior,Evidence,PosteriorMean,PosteriorMode,CredibleInterval,PosteriorPredictive,BayesFactor,HierarchicalInference,LatentVariable,AnalyticInference,ApproximateInference,MCMC,MetropolisHastings,GibbsSampling,HamiltonianMonteCarlo,NUTS,EffectiveSampleSize,RHat,BayesAction,PosteriorPredictiveCheck}
CoreRelations := {infers,hasPrior,hasLikelihood,updatesTo,normalisedBy,hasPosterior,predicts,hasCredibleInterval,comparesHypothesis,hasLatentVariable,sampledBy,diagnosedBy,evaluatedBy,optimises,conditionedOn,constrainedBy,equivalentTo,tracesTo}

TypedRelations := BayesianInference--infers-->Parameter ∧ Parameter--hasPrior-->Prior ∧ Model--hasLikelihood-->Likelihood ∧ PriorAndLikelihood--updatesTo-->Posterior ∧ Posterior--normalisedBy-->Evidence ∧ Posterior--predicts-->FutureData ∧ Parameter--hasCredibleInterval-->CredibleInterval ∧ BayesianInference--comparesHypothesis-->BayesFactor ∧ Model--hasLatentVariable-->LatentVariable ∧ Posterior--sampledBy-->MCMC ∧ MCMC--diagnosedBy-->ConvergenceDiagnostic ∧ Model--evaluatedBy-->PosteriorPredictiveCheck ∧ BayesAction--optimises-->PosteriorExpectedLoss ∧ BayesianInference--tracesTo-->Provenance

InferenceRules := {Prior∧Likelihood⇒Posterior,Posterior∧FutureLikelihood⇒PosteriorPredictive,ConjugatePrior∧CompatibleLikelihood⇒ClosedFormPosterior,SquaredErrorLoss⇒PosteriorMeanOptimal,AbsoluteErrorLoss⇒PosteriorMedianOptimal,RHatHigh∨ESSLow∨Divergence⇒FlagComputationalRisk,PosteriorPredictiveFailure⇒FlagModelAdequacyRisk,PriorSensitivityHigh⇒FlagPriorDependence,BayesianAssociation↛CausalEffect}

CanonicalBayesianInferenceRepresentation := ParseInferenceRequest→ResolveModelParametersAndData→SpecifyLikelihood→SpecifyPrior→RunPriorPredictiveChecks→ComputePosterior→SelectAnalyticApproximateOrSamplingMethod→DiagnoseComputation→ComputePosteriorSummariesAndDerivedQuantities→GenerateCredibleIntervals→GeneratePosteriorPredictiveDistribution→RunPosteriorPredictiveChecks→AssessSensitivity→EvaluatePredictivePerformance→ResolveDecisionTarget→ValidateBayesianInference

BayesianInferenceGraph := G=(V,E)
V := {BayesianInferences,Parameters,Priors,Likelihoods,Posteriors,Evidences,PosteriorMeans,PosteriorModes,CredibleIntervals,PosteriorPredictives,BayesFactors,HierarchicalInferences,LatentVariables,AnalyticInferences,ApproximateInferences,MCMCChains,MetropolisHastingsSamplers,GibbsSamplers,HamiltonianMonteCarloSamplers,NUTSSamplers,EffectiveSampleSizes,RHatDiagnostics,BayesActions,PosteriorPredictiveChecks,EvidenceNodes,ProvenanceNodes}
E := {infers,hasPrior,hasLikelihood,updatesTo,normalisedBy,hasPosterior,predicts,hasCredibleInterval,comparesHypothesis,hasLatentVariable,sampledBy,diagnosedBy,evaluatedBy,optimises,conditionedOn,constrainedBy,equivalentTo,tracesTo}

CONFIDENCE(BayesianInferenceResolution) := weighted_mean(ModelResolution,ParameterResolution,PriorValidity,LikelihoodValidity,PosteriorPropriety,IdentifiabilityValidity,ComputationalValidity,ConvergenceQuality,PriorPredictiveQuality,PosteriorPredictiveQuality,SensitivityAssessment,PredictiveValidation,ProvenanceQuality)

AI_QUERY_INTERPRETATION := Query→DetectBayesianInferenceIntent→IdentifyModelParametersDataAndPriorContext→ResolveEstimationPredictionHypothesisDecisionOrModelComparisonIntent→IdentifyRequestedPosteriorSummaryCredibleIntervalBayesFactorOrDerivedQuantity→ResolveComputationalSensitivityAndValidationRequirements

AI_REASONING_FLOW := Query→ParseBayesianInferenceProblem→ResolveModelParametersAndData→SpecifyLikelihood→SpecifyPrior→ExecutePriorPredictiveReasoning→ComputePosterior→SelectInferenceMethod→DiagnoseComputation→ComputePosteriorSummariesAndDerivedQuantities→GenerateCredibleIntervals→GeneratePosteriorPredictiveDistribution→ExecutePosteriorPredictiveChecks→AssessPriorLikelihoodAndModelSensitivity→EvaluatePredictivePerformance→ResolveDecisionOrScientificInterpretation→UpdateConfidence→PreserveProvenance→GenerateExplanation

AI_CONSUMPTION := {ParseBayesianInferenceEntities,ParseTypedRelations,ResolveModels,ResolveParameters,ResolveParameterSpaces,ResolveLikelihoods,ResolvePriors,ComputeBayesRule,ComputeMarginalLikelihoods,ComputePosteriors,ComputePosteriorMeans,ComputePosteriorMedians,ComputePosteriorModes,ComputePosteriorVariances,ComputeCredibleIntervals,ComputeHighestDensityIntervals,ComputeBayesFactors,GeneratePriorPredictiveSamples,GeneratePosteriorPredictiveSamples,ResolveHierarchicalInference,ResolveLatentVariables,ExecuteGridInference,ExecuteLaplaceInference,ExecuteVariationalInference,ExecuteImportanceSampling,ExecuteMetropolisHastings,ExecuteGibbsSampling,ExecuteHamiltonianMonteCarlo,ExecuteNUTS,ComputeEffectiveSampleSize,ComputeRHat,ComputeMonteCarloStandardError,DetectDivergences,ExecutePriorSensitivityAnalysis,ExecuteLikelihoodSensitivityAnalysis,ExecutePosteriorPredictiveChecks,ExecutePSISLOO,ComputeWAIC,ResolveBayesActions,ValidateBayesianInferenceSemantics,CompareSemanticEquivalence,UpdateConfidence,PreserveProvenance,GenerateFormalExplanation}

UPDATE_MODEL := NewDataPriorLikelihoodModelOrEvidence→ValidateInput→IdentifyAffectedInferenceClaims→UpdateBayesianInferenceState→RecomputePriorPredictiveIfRequired→RecomputePosterior→RerunComputationalDiagnostics→RecomputePosteriorSummariesDerivedQuantitiesAndPredictiveDistribution→RerunSensitivityAndPredictiveChecks→RecomputeConfidence→PreserveProvenance→VersionModel
DataUpdate := NewObservedData⇒recompute(Likelihood,Posterior,PosteriorSummaries,PosteriorPredictive)
PriorUpdate := ChangedPrior⇒recompute(PriorPredictive,Posterior,CredibleIntervals,SensitivityStatus)
LikelihoodUpdate := ChangedLikelihood⇒reevaluate(Posterior,PosteriorPredictiveChecks,PredictivePerformance)
InferenceMethodUpdate := ChangedComputation⇒reevaluate(ConvergenceDiagnostics,MonteCarloError,ApproximationQuality)

FLOW_HASH_INPUT := canonical(ONTOLOGY,CoreClasses,CoreRelations,InferenceRules,BayesianInference,BayesianInferenceValidation,BayesianInferenceConstraint,SEMANTIC_EQUIVALENCE,AI_REASONING_FLOW)
FLOW_HASH := HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {BayesianInferenceEntityResolution,Model,Parameters,ParameterSpace,Prior,Likelihood,Posterior,Evidence,PosteriorMean,PosteriorMedian,PosteriorMode,PosteriorVariance,CredibleIntervals,PosteriorPredictive,BayesFactors,LatentVariables,MCMCChains,RHat,EffectiveSampleSize,MonteCarloStandardError,Divergences,PriorPredictiveStatus,PosteriorPredictiveStatus,SensitivityStatus,PredictivePerformance,DecisionTarget,ValidationStatus,Confidence,Provenance,Explanation}

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA