Posterior Probability Ontology

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

BEGIN_FLOW

PriorProbability := PreDataProbabilityAssignment∧BayesianModelComponent∧UncertaintyRepresentation∧InformationEncoding∧RegularisationMechanism
Parameter := θ
ParameterSpace := Θ
Hypothesis := H
Model := M
ObservedData := D
PriorProbability := P(H)
PriorDensity := p(θ)
Likelihood := p(D|θ)
Posterior := p(θ|D)
BayesianUpdate := Prior∧Likelihood→Posterior
BayesRule := p(θ|D)=p(D|θ)p(θ)/p(D)
PriorRole := {RepresentPreDataUncertainty,EncodeExistingKnowledge,ConstrainParameterSpace,RegulariseInference,SupportPrediction,EnableBayesianUpdating}
PriorInformation := InformationAvailableBeforeCurrentLikelihoodContribution
PriorOdds := P(H1)/P(H0)
PriorModelProbability := P(M)
PriorParameterDistribution := p(θ)
DiscretePrior := Σ_θp(θ)=1
ContinuousPrior := ∫_Θp(θ)dθ=1
ProperPrior := NormalisableProbabilityDistribution
ImproperPrior := NonNormalisablePriorMeasure
ImproperPriorConstraint := ImproperPriorPermittedOnlyIfPosteriorProper
PosteriorPropriety := ∫_Θp(D|θ)p(θ)dθ<∞
InformativePrior := PriorConcentratedUsingSubstantivePreDataInformation
WeaklyInformativePrior := PriorBroadEnoughToPermitLearning∧RestrictiveEnoughToExcludeImplausibleExtremes
DiffusePrior := BroadPriorWithLowConcentrationAcrossLargeParameterRegion
ReferencePrior := PriorConstructedUnderFormalLimitedInformationCriterion
SubjectivePrior := PriorReflectingExplicitExpertOrScientificJudgement
EmpiricalBayesPrior := PriorHyperparametersEstimatedFromObservedOrRelatedData
HierarchicalPrior := HyperparameterControlsParameterPrior
Hyperparameter := η
Hyperprior := p(η)
HierarchicalPriorStructure := η~p(η)∧θ|η~p(θ|η)
ExchangeablePrior := JointPriorInvariantUnderPermutationOfExchangeableUnits
IndependentPrior := p(θ1,...,θk)=Π_jp(θj)
DependentPrior := JointPriorEncodesDependenceAcrossParameters
JointPrior := p(θ1,...,θk)
PriorSupport := {θ:p(θ)>0}
SupportConstraint := ScientificallyPlausibleParameterValuesShouldNotBeExcludedWithoutJustification
HardConstraintPrior := PriorAssignsZeroProbabilityOutsideAllowedRegion
SoftConstraintPrior := PriorDownweightsImplausibleValuesWithoutAssigningZeroProbability
RegularisingPrior := PriorDesignedToStabiliseEstimation
ShrinkagePrior := PriorEncouragingParametersTowardReferenceValueOrSharedStructure
GaussianPrior := θ~Normal(μ0,σ0²)
BetaPrior := p~Beta(α,β)
GammaPrior := λ~Gamma(α,β)
DirichletPrior := π~Dirichlet(α1,...,αK)
HalfNormalPrior := σ~HalfNormal(s)
HalfTPrior := σ~HalfT(ν,s)
LaplacePrior := θ~Laplace(μ,b)
UniformPrior := θ~Uniform(a,b)
LogNormalPrior := θ~LogNormal(μ,σ²)
PriorMean := E[θ]
PriorVariance := Var(θ)
PriorMedian := Median(θ)
PriorMode := argmax_θp(θ)
PriorQuantile := Q_q(θ)
PriorPrecision := 1/PriorVariance for compatible scalar distributions
PriorConcentration := DegreeOfProbabilityMassConcentration
PriorScale := ParameterControllingPriorDispersion
ConjugatePrior := PriorFamilyYieldingPosteriorInSameFamilyUnderCompatibleLikelihood
BetaBinomialPrior := p~Beta(α,β)
BetaBinomialPosterior := p|D~Beta(α+x,β+n-x)
GammaPoissonPrior := λ~Gamma(α,β)
NormalNormalPrior := θ~Normal(μ0,τ0²)
ConjugacyBenefit := AnalyticPosterior∧EfficientSequentialUpdating
EffectivePriorSampleSize := MeasureOfPriorInformationRelativeToLikelihoodUnderModelSpecificDefinition
PriorPredictiveDistribution := p(D)=∫p(D|θ)p(θ)dθ
PriorPredictiveSimulation := θ~p(θ)→D_rep~p(D_rep|θ)
PriorPredictiveCheck := SimulatePriorPredictive→CompareWithScientificallyPlausibleDataRangeAndStructure
PriorPredictiveFailure := PriorAndLikelihoodGenerateImplausibleDataWithSubstantialProbability
PriorElicitation := DomainKnowledge→ProbabilityJudgements→PriorParameters
ExpertElicitation := StructuredExpertJudgement→PriorDistribution
QuantileElicitation := SpecifyPriorQuantiles→FitDistributionParameters
MomentElicitation := SpecifyPriorMeanAndVariance→ResolvePriorFamilyParameters
ProbabilityElicitation := SpecifyProbabilityOfParameterRegion→ConstrainPrior
Transformation := φ=g(θ)
TransformedPrior := p_φ(φ)=p_θ(g^(-1)(φ))|d g^(-1)(φ)/dφ|
ParameterisationDependence := ApparentlyDiffusePriorOnOneScaleMayBeInformativeOnAnotherScale
PriorOnLogScale := log(θ)~Distribution for positive θ
PriorOnLogitScale := logit(p)~Distribution for probability p
PriorSensitivity := PosteriorChangesUnderAlternativeReasonablePriors
PriorSensitivityAnalysis := DefineReasonablePriorSet→FitPosteriorForEachPrior→ComparePosteriorTargets
RobustPriorConclusion := ScientificConclusionStableAcrossReasonablePriorChoices
PriorDominance := PosteriorStronglyInfluencedByPriorRelativeToLikelihood
LikelihoodDominance := DataLikelihoodOverwhelmsPriorForTargetParameter
PriorDataConflict := PriorPlacesMassWhereLikelihoodHasLowSupport
PriorPosteriorComparison := PriorDistribution versus PosteriorDistribution
BayesianLearning := PosteriorDiffersFromPriorDueToLikelihoodInformation
InformationGain := Divergence(Posterior,Prior)
KLDivergence := KL(posterior||prior)
PriorEntropy := -∫p(θ)logp(θ)dθ
ModelPrior := P(M_k)
ModelPriorConstraint := Σ_kP(M_k)=1
PosteriorModelProbability := P(M_k|D)∝p(D|M_k)P(M_k)
BayesFactorRelation := PosteriorModelOdds=BayesFactor×PriorModelOdds
BayesFactorPriorSensitivity := MarginalLikelihoodCanBeHighlySensitiveToParameterPriorScale
HypothesisPrior := P(H0),P(H1)
HypothesisPriorConstraint := P(H0)+P(H1)=1 for exhaustive binary hypotheses
SpikeAndSlabPrior := PointMassAtNull∧ContinuousAlternativeDistribution
VariableSelectionPrior := PriorOnModelInclusionIndicators
InclusionProbability := P(γ_j=1)
PosteriorInclusionProbability := P(γ_j=1|D)
SparsityPrior := PriorEncouragingManyParametersNearOrAtZero
RidgePriorRelation := GaussianCoefficientPrior↔L2RegularisationUnderMAP
LassoPriorRelation := LaplaceCoefficientPrior↔L1RegularisationUnderMAP
HierarchicalShrinkagePrior := GlobalScale∧LocalScales→AdaptiveShrinkage
PriorForProbability := DistributionSupportedOn[0,1]
PriorForPositiveParameter := PositiveSupportDistribution
PriorForUnboundedParameter := RealLineDistribution
SupportMatchingRule := PriorSupportMustMatchParameterConstraints
PriorIndependenceAssumption := ProductPriorImpliesPriorIndependenceNotNecessarilyPosteriorIndependence
PosteriorDependence := LikelihoodCanInduceDependenceBetweenParametersEvenUnderIndependentPriors
PriorMisspecification := PriorPoorlyRepresentsPlausibleParameterValuesOrKnowledge
OverlyNarrowPrior := ExcessiveConcentration→PotentialPosteriorBiasIfPriorIncorrect
OverlyWidePrior := ExtremeDispersion→PotentialComputationalAndPredictiveProblems
PriorAndIdentifiability := InformativePriorCanRegulariseWeakIdentificationButCannotCreateDataInformation
PriorTransparency := ReportPriorFamilyParametersScaleRationaleAndSensitivity
PriorReproducibility := PriorSpecificationMustBeExplicitEnoughToReconstruct
PriorProvenance := SourceOfPriorInformation∧ElicitationMethod∧Transformation∧Version
DoubleUseOfData := SameDataUsedToConstructPriorAndLikelihoodWithoutAccounting→InterpretationRisk
ExternalDataPrior := HistoricalOrExternalDataset→PriorConstruction
PowerPrior := HistoricalLikelihoodRaisedToWeight→PriorContribution
HistoricalBorrowingConstraint := DegreeOfBorrowingMustBeExplicitAndSensitivityAssessed
PriorProbabilityDoesNotEqualPosteriorProbability := P(H)≠P(H|D) unless data provide no update
PriorProbabilityDoesNotImplyTruth := ProbabilityAssignmentRepresentsModelledUncertaintyNotGuaranteedTruth

PriorConstraint := ConstraintOn(ParameterSpace,Support,PriorFamily,Hyperparameters,InformationSource,Parameterisation,Propriety,ComputationalBehaviour)
PriorConsistency := ParameterSemantics∧Support∧DistributionFamily∧Hyperparameters∧InformationSource jointly_compatible
InvalidPrior := SupportMismatch∨UndefinedHyperparameters∨UnjustifiedZeroProbabilityForPlausibleRegion∨ImproperPosterior∨HiddenDataReuse∨SeverePriorPredictiveImplausibilityUnaddressed
PriorValidation := ResolveParameterAndScale→ResolveScientificConstraints→IdentifyPriorInformationSource→SelectPriorFamily→SetHyperparameters→ValidateSupport→RunPriorPredictiveSimulation→AssessPriorPredictivePlausibility→AssessComputationalGeometry→PerformPriorSensitivityAnalysis→RecordPriorProvenance→ValidateResult
PriorProbabilityState := {Parameter,ParameterSpace,PriorFamily,Hyperparameters,PriorMean,PriorVariance,PriorSupport,PriorPredictiveDistribution,PriorPredictiveStatus,InformationSource,ElicitationMethod,Parameterisation,ProprietyStatus,SensitivityStatus,DataConflictStatus,Evidence,Confidence,Provenance}
SEMANTIC_EQUIVALENCE(PriorA,PriorB) := same(ParameterSemantics)∧same(ParameterisationOrValidTransformation)∧same(Support)∧same(ProbabilityMeasureAfterTransformation)∧same(InformationContent)

PriorProbabilityOntology := Classes∧Relations∧Constraints∧Axioms∧InferenceRules
CoreClasses := {PriorProbability,PriorDistribution,InformativePrior,WeaklyInformativePrior,DiffusePrior,ImproperPrior,ConjugatePrior,HierarchicalPrior,Hyperprior,RegularisingPrior,ShrinkagePrior,PriorPredictiveDistribution,PriorPredictiveCheck,PriorElicitation,PriorSensitivity,PriorDataConflict,ModelPrior,HypothesisPrior,SpikeAndSlabPrior,PriorProvenance}
CoreRelations := {priorFor,hasSupport,hasHyperparameter,derivedFrom,updatedBy,produces,checkedBy,elicitedFrom,transformedBy,regularises,comparedWith,conflictsWith,sensitiveTo,constrainedBy,equivalentTo,tracesTo}
TypedRelations := PriorDistribution--priorFor-->Parameter ∧ PriorDistribution--hasSupport-->ParameterSpace ∧ HierarchicalPrior--hasHyperparameter-->Hyperparameter ∧ PriorDistribution--derivedFrom-->PriorInformation ∧ PriorDistribution--updatedBy-->Likelihood ∧ PriorDistribution--produces-->PriorPredictiveDistribution ∧ PriorPredictiveDistribution--checkedBy-->PriorPredictiveCheck ∧ PriorDistribution--elicitedFrom-->ExpertKnowledge ∧ RegularisingPrior--regularises-->ParameterEstimate ∧ PriorDistribution--comparedWith-->PosteriorDistribution ∧ PriorDistribution--conflictsWith-->Likelihood ∧ PriorDistribution--tracesTo-->PriorProvenance
InferenceRules := {ValidPrior∧Likelihood⇒PosteriorCanBeComputedIfEvidenceFinite,ProperPrior⇒PriorProbabilityMassNormalised,ImproperPrior∧FiniteEvidence⇒PosteriorMayBeProper,SupportMismatch⇒FlagPriorInvalid,PriorPredictiveImplausibility⇒FlagPriorSpecificationRisk,PosteriorSensitiveToReasonablePriorChanges⇒FlagPriorDependence,PriorDataConflict⇒RequireSensitivityAndModelAssessment,GaussianPriorOnCoefficients∧MAP⇒RidgeLikePenaltyUnderCompatibleModel,LaplacePriorOnCoefficients∧MAP⇒LassoLikePenaltyUnderCompatibleModel}
CanonicalPriorProbabilityRepresentation := ParsePriorRequest→ResolveParameterAndParameterSpace→ResolveInformationSource→ResolveParameterisation→SelectPriorFamily→ResolveHyperparameters→ValidateSupportAndPropriety→GeneratePriorPredictiveDistribution→ExecutePriorPredictiveChecks→AssessPriorDataConflict→ExecutePriorSensitivityAnalysis→PreservePriorProvenance→ValidatePriorProbability
PriorProbabilityGraph := G=(V,E)
V := {PriorProbabilities,PriorDistributions,InformativePriors,WeaklyInformativePriors,DiffusePriors,ImproperPriors,ConjugatePriors,HierarchicalPriors,Hyperpriors,RegularisingPriors,ShrinkagePriors,PriorPredictiveDistributions,PriorPredictiveChecks,PriorElicitations,PriorSensitivities,PriorDataConflicts,ModelPriors,HypothesisPriors,SpikeAndSlabPriors,Parameters,ParameterSpaces,Likelihoods,Posteriors,PriorProvenanceNodes,EvidenceNodes}
E := {priorFor,hasSupport,hasHyperparameter,derivedFrom,updatedBy,produces,checkedBy,elicitedFrom,transformedBy,regularises,comparedWith,conflictsWith,sensitiveTo,constrainedBy,equivalentTo,tracesTo}
CONFIDENCE(PriorProbabilityResolution) := weighted_mean(ParameterResolution,ParameterSpaceValidity,PriorFamilyValidity,HyperparameterValidity,SupportValidity,InformationSourceQuality,PriorPredictiveQuality,ProprietyValidity,SensitivityAssessment,ProvenanceQuality)
AI_QUERY_INTERPRETATION := Query→DetectPriorProbabilityIntent→IdentifyParameterHypothesisOrModel→ResolvePriorFamilySupportAndHyperparameters→IdentifyInformationSourceAndElicitationContext→ResolvePriorPredictiveSensitivityOrModelComparisonIntent
AI_REASONING_FLOW := Query→ParsePriorProbabilityProblem→ResolveParameterAndScale→ResolveParameterSpace→ResolveInformationSource→SelectPriorFamily→ResolveHyperparameters→ValidateSupport→ValidatePropriety→GeneratePriorPredictiveDistribution→ExecutePriorPredictiveChecks→AssessPriorDataConflict→PerformSensitivityAnalysis→ValidateParameterisation→UpdateConfidence→PreserveProvenance→GenerateExplanation
AI_CONSUMPTION := {ParsePriorProbabilityEntities,ParseTypedRelations,ResolveParameters,ResolveParameterSpaces,ResolvePriorFamilies,ResolveHyperparameters,ValidatePriorSupport,ValidatePriorPropriety,ComputePriorMeans,ComputePriorVariances,ComputePriorQuantiles,GeneratePriorPredictiveSamples,ExecutePriorPredictiveChecks,ExecutePriorElicitation,TransformPriors,ComputeModelPriorOdds,ComputeHypothesisPriorOdds,ResolveConjugatePriors,ResolveHierarchicalPriors,ResolveShrinkagePriors,DiagnosePriorDataConflict,ExecutePriorSensitivityAnalysis,ComputeInformationGain,ValidatePriorProbabilitySemantics,CompareSemanticEquivalence,UpdateConfidence,PreserveProvenance,GenerateFormalExplanation}
UPDATE_MODEL := NewPriorInformationHyperparameterDataOrEvidence→ValidateInput→IdentifyAffectedPriorClaims→UpdatePriorProbabilityState→RecomputePriorDistribution→RevalidateSupportAndPropriety→RegeneratePriorPredictiveDistribution→RerunPriorPredictiveChecks→ReassessPriorDataConflictAndSensitivity→RecomputeConfidence→PreserveProvenance→VersionModel
InformationUpdate := NewPriorInformation⇒recompute(PriorFamilyOrHyperparameters,PriorPredictiveDistribution,SensitivityStatus)
HyperparameterUpdate := ChangedHyperparameters⇒recompute(PriorMoments,PriorQuantiles,PriorPredictiveDistribution,PosteriorInfluence)
ParameterisationUpdate := ChangedParameterScale⇒transform(PriorDensity,Support,Hyperparameters)
FLOW_HASH_INPUT := canonical(ONTOLOGY,CoreClasses,CoreRelations,InferenceRules,PriorProbability,PriorValidation,PriorConstraint,SEMANTIC_EQUIVALENCE,AI_REASONING_FLOW)
FLOW_HASH := HASH(FLOW_HASH_INPUT)
OUTPUT_MODEL := {PriorProbabilityEntityResolution,Parameter,ParameterSpace,PriorFamily,Hyperparameters,PriorMean,PriorVariance,PriorSupport,PriorPredictiveDistribution,PriorPredictiveStatus,InformationSource,ElicitationMethod,Parameterisation,ProprietyStatus,SensitivityStatus,DataConflictStatus,ValidationStatus,Confidence,Provenance,Explanation}

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA