Bayesian Computation Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Bayesian Computation Ontology within the Kireth Research Semantic Library. Bayesian computation concerns the numerical and algorithmic evaluation of posterior distributions, predictive quantities, marginal likelihoods and Bayesian decision quantities when analytical solutions are unavailable or impractical. This ontology supports Bayes' theorem, prior and likelihood construction, posterior computation, conjugate and nonconjugate models, MCMC, importance sampling, sequential Monte Carlo, variational inference, Laplace approximation, approximate Bayesian computation, posterior predictive checking, marginal likelihood estimation, model comparison, hierarchical modelling, numerical stability, diagnostics, scalable computation, verification and reproducibility.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Bayesian Computation Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Computational Mathematics"
 SPECIALISATION = "Computational Statistics / Bayesian Computation"
 CANONICAL_PAGE = "https://kirethframeworkold.co.uk/books/semantic-library/ai-core/ontologies/mathematical-ontologies/computational-mathematics-ontologies/computational-statistics/bayesian-computation-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = BayesianComputation
 ENTITY_CLASS = PosteriorInferenceComputationFramework
 PRIMARY_CONTEXT = ComputationalStatistics
 DEFINITION = "Bayesian computation is the numerical or algorithmic evaluation and approximation of posterior distributions, posterior expectations, predictive distributions and related Bayesian quantities."
 SYMBOLIC_FORMS = [
  "p(theta|y) = p(y|theta)p(theta)/p(y)",
  "p(y_new|y)=integral p(y_new|theta)p(theta|y)dtheta",
  "E[g(theta)|y]=integral g(theta)p(theta|y)dtheta",
  "p(y)=integral p(y|theta)p(theta)dtheta"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for priors, likelihoods, posterior distributions, posterior approximation, Bayesian sampling, predictive inference, model evidence, diagnostics, numerical stability and reproducibility."
}

SCOPE {
 INCLUDE = [
  BayesianComputation,BayesianInference,
  BayesianStatistics,PosteriorComputation,
  BayesTheorem,BayesRule,
  PriorDistribution,PriorDensity,
  PriorProbability,PriorPredictiveDistribution,
  Likelihood,SamplingModel,
  DataLikelihood,LogLikelihood,
  PosteriorDistribution,PosteriorDensity,
  PosteriorProbability,LogPosterior,
  NormalisingConstant,Evidence,
  MarginalLikelihood,
  PosteriorPredictiveDistribution,
  PriorPredictiveCheck,
  PosteriorPredictiveCheck,PPC,
  PredictiveDistribution,
  PredictiveProbability,
  BayesianEstimator,
  PosteriorMean,PosteriorMedian,
  PosteriorMode,MAP,
  MaximumAPosteriori,
  PosteriorVariance,
  PosteriorStandardDeviation,
  CredibleInterval,
  HighestPosteriorDensityInterval,HPD,
  PosteriorQuantile,
  BayesianDecisionTheoryRelationship,
  UtilityFunction,LossFunction,
  BayesEstimatorRelationship,
  ConjugatePrior,Conjugacy,
  NonconjugateModel,
  ExponentialFamilyRelationship,
  BetaBinomialModel,
  GammaPoissonModel,
  NormalNormalModel,
  NormalInverseGammaRelationship,
  HierarchicalModel,
  MultilevelModel,
  Hyperparameter,Hyperprior,
  PartialPooling,CompletePooling,
  NoPooling,
  Exchangeability,
  ConditionalIndependence,
  GraphicalModelRelationship,
  DirectedAcyclicGraphRelationship,
  LatentVariable,
  MissingData,
  DataAugmentation,
  GibbsSampling,
  MarkovChainMonteCarlo,MCMC,
  MetropolisHastings,
  HamiltonianMonteCarlo,HMC,
  NoUTurnSampler,NUTS,
  SliceSampling,
  AdaptiveMCMC,
  SequentialMonteCarlo,SMC,
  ParticleFilter,
  ImportanceSampling,
  SelfNormalisedImportanceSampling,
  RejectionSampling,
  AnnealedImportanceSampling,AIS,
  BridgeSampling,
  ThermodynamicIntegration,
  NestedSamplingRelationship,
  VariationalInference,VI,
  MeanFieldVariationalInference,
  StructuredVariationalInference,
  EvidenceLowerBound,ELBO,
  KLdivergence,
  ReverseKLDivergenceRelationship,
  ForwardKLDivergenceRelationship,
  LaplaceApproximation,
  GaussianApproximation,
  ExpectationPropagationRelationship,
  IntegratedNestedLaplaceApproximation,INLARelationship,
  ApproximateBayesianComputation,ABC,
  LikelihoodFreeInference,
  SyntheticLikelihood,
  SimulationBasedInference,
  NeuralPosteriorEstimationRelationship,
  NeuralLikelihoodEstimationRelationship,
  NeuralRatioEstimationRelationship,
  PosteriorApproximation,
  ExactInferenceRelationship,
  ApproximateInference,
  DeterministicApproximation,
  StochasticApproximation,
  NumericalIntegration,
  Quadrature,
  AdaptiveQuadrature,
  GaussianQuadrature,
  MonteCarloIntegration,
  QuasiMonteCarlo,
  GridApproximation,
  PosteriorGrid,
  ImportanceWeight,
  EffectiveSampleSize,ESS,
  MonteCarloStandardError,MCSE,
  Autocorrelation,
  IntegratedAutocorrelationTime,
  RHat,
  GelmanRubinStatistic,
  DivergentTransition,
  TreeDepth,
  AcceptanceRate,
  Mixing,
  Convergence,
  PosteriorConvergence,
  VariationalConvergence,
  OptimisationConvergence,
  ELBOConvergence,
  PosteriorCalibration,
  SimulationBasedCalibration,SBC,
  Calibration,
  CoverageProbability,
  PriorSensitivity,
  SensitivityAnalysis,
  RobustBayesRelationship,
  WeaklyInformativePrior,
  InformativePrior,
  NoninformativePriorRelationship,
  ReferencePriorRelationship,
  ImproperPrior,
  ProperPrior,
  PriorPredictiveSimulation,
  PosteriorPredictiveSimulation,
  BayesianModelComparison,
  ModelEvidence,
  BayesFactor,
  PosteriorModelProbability,
  ModelAveraging,
  BayesianModelAveraging,BMA,
  WAICRelationship,
  LeaveOneOutCrossValidationRelationship,
  LOO,
  PredictiveModelComparison,
  InformationCriterionRelationship,
  Marginalisation,
  NuisanceParameter,
  RaoBlackwellisation,
  CollapsedSampling,
  Reparameterisation,
  CentredParameterisation,
  NonCentredParameterisation,
  WhiteningTransformation,
  TransformToUnconstrainedSpace,
  JacobianAdjustment,
  CholeskyParameterisation,
  CorrelationMatrix,
  CovarianceMatrix,
  PrecisionMatrix,
  LinearAlgebra,
  MatrixComputation,
  CholeskyDecomposition,
  QRDecomposition,
  SingularValueDecomposition,SVD,
  SparseMatrix,
  GaussianProcessRelationship,
  StateSpaceModelRelationship,
  HiddenMarkovModelRelationship,
  MixtureModel,
  BayesianRegression,
  BayesianLinearRegression,
  BayesianLogisticRegression,
  BayesianGLM,
  BayesianHierarchicalRegression,
  BayesianTimeSeries,
  BayesianSurvivalModelRelationship,
  BayesianNonparametricsRelationship,
  DirichletProcessRelationship,
  GaussianProcessBayesRelationship,
  RandomNumberGeneration,RNG,
  PRNG,RandomSeed,RandomStream,
  ParallelRandomNumberGeneration,
  NumericalStability,
  FloatingPointArithmetic,
  MachinePrecision,RoundOffError,
  Underflow,Overflow,
  LogSumExp,
  StableLogDensity,
  StableLikelihood,
  AutomaticDifferentiationRelationship,
  GradientOfLogPosterior,
  HessianOfLogPosterior,
  FisherInformationRelationship,
  ComputationalComplexity,
  TimeComplexity,SpaceComplexity,
  CostPerSample,
  CostPerGradient,
  PosteriorEvaluationCost,
  ParallelBayesianComputation,
  DistributedBayesianComputation,
  GPUBayesianComputation,
  VectorisedBayesianComputation,
  ChainParallelism,
  WithinChainParallelism,
  DataParallelLikelihood,
  HighPerformanceComputing,
  ParallelComputing,
  DistributedComputing,
  GPUComputing,
  Verification,CodeVerification,
  KnownPosteriorTest,
  ConjugateModelTest,
  SimulationStudy,
  SimulatedTruth,
  Benchmark,
  Validation,
  Reproducibility,Repeatability,
  Provenance,Configuration,
  SoftwareVersion,LibraryVersion,
  SamplerVersion,SeedMetadata
 ]
 EXCLUDE_AS_PRIMARY = [
  FrequentistInferenceWithoutPosteriorSemantics,
  PureSymbolicBayesianDerivationWithoutComputationalExecution,
  GenericMonteCarloWithoutBayesianTargetQuantities,
  MachineLearningWithoutBayesianProbabilityOrPosteriorSemantics
 ]
 DISAMBIGUATION_RULE = "When priors and likelihoods are combined computationally to obtain posterior or posterior-predictive quantities, interpret the context as Bayesian computation."
}

CORE_CONCEPTS {
 BayesianComputation = "Numerical computation of posterior and predictive Bayesian quantities."
 PriorDistribution = "Probability distribution representing uncertainty before conditioning on observed data."
 Likelihood = "Probability model for observed data as a function of unknown parameters."
 PosteriorDistribution = "Updated probability distribution after conditioning on observed data."
 MarginalLikelihood = "Probability of observed data under a model after integrating over parameters."
 PosteriorPredictiveDistribution = "Distribution of future or replicated data conditional on observed data."
 BayesianModelComparison = "Comparison of models using posterior predictive or evidence-based quantities."
 PosteriorApproximation = "Numerical or stochastic approximation to an analytically intractable posterior."
}

RELATIONS {
 BayesianComputation --is_child_of--> ComputationalStatistics
 BayesianComputation --uses--> BayesTheorem
 BayesTheorem --combines--> PriorAndLikelihood
 PriorDistribution --updates_to--> PosteriorDistribution
 Likelihood --contributes_to--> PosteriorDistribution
 MarginalLikelihood --normalises--> PosteriorDistribution
 PosteriorDistribution --induces--> PosteriorPredictiveDistribution
 MCMC --samples_from--> PosteriorDistribution
 ImportanceSampling --approximates--> PosteriorExpectation
 SequentialMonteCarlo --approximates--> PosteriorSequence
 VariationalInference --approximates--> PosteriorDistribution
 LaplaceApproximation --approximates--> PosteriorLocally
 ABC --approximates--> PosteriorWithoutExplicitLikelihood
 BayesFactor --compares--> MarginalLikelihoods
 BayesianModelAveraging --weights_by--> PosteriorModelProbability
 Verification --checks--> BayesianComputationImplementation
 Reproducibility --requires--> ModelSeedAndVersionMetadata
}

OPERATORS {
 POSTERIOR(prior,likelihood,data) := normalise(likelihood(data|theta)*prior(theta))
 LOG_POSTERIOR(theta,data) := log_likelihood(data|theta)+log_prior(theta)+constant
 POSTERIOR_EXPECTATION(g,samples) := mean({g(theta_i)})
 POSTERIOR_PREDICTIVE(theta_samples) := draw_y_new_from_p_y_given_theta(theta_samples)
 BAYES_FACTOR(M1,M2) := marginal_likelihood(M1)/marginal_likelihood(M2)
 IMPORTANCE_POSTERIOR_ESTIMATE(g,w,theta) := sum_i w_i*g(theta_i)/sum_i w_i
 ELBO(q) := E_q[log p(data,theta)-log q(theta)]
 MAP_ESTIMATE := argmax_theta LOG_POSTERIOR(theta,data)
}

AXIOMS {
 A1 = "Bayesian computation combines prior information and likelihood information through Bayes' theorem."
 A2 = "Posterior normalisation can be analytically intractable even when unnormalised density evaluation is possible."
 A3 = "Sampling, quadrature and optimisation methods approximate different posterior quantities in different ways."
 A4 = "Posterior predictive checking assesses implications of the fitted Bayesian model."
 A5 = "Marginal likelihood computation is generally more demanding than posterior-mode estimation."
 A6 = "Approximate inference must preserve the distinction between target posterior and approximation."
 A7 = "Prior choice can materially affect posterior inference, especially with weak data."
 A8 = "Bayesian-computation results require explicit prior, likelihood, posterior, approximation, diagnostics and reproducibility semantics."
}

REASONING {
 R1_CONTEXT = identify(data,model,parameters,latent_variables,prior_information,target_quantities,accuracy_goal,budget)
 R2_PRIOR = define(proper_or_improper_prior_hyperpriors_and_prior_predictive_implications)
 R3_LIKELIHOOD = define(data_model_conditional_independence_and_stable_log_likelihood)
 R4_POSTERIOR = derive_unnormalised_log_posterior_and_parameter_constraints)
 R5_METHOD = choose(conjugate,quadrature,MCMC,SMC,importance,variational,Laplace,ABC_or_hybrid_method)
 R6_PARAMETERISATION = choose(centred,noncentred,transformed,whitened_or_Cholesky_parameterisation)
 R7_EXECUTION = run_sampler_optimiser_quadrature_or_variational_algorithm)
 R8_DIAGNOSTICS = assess(ESS,MCSE,RHat,divergences,ELBO,importance_weights_or_quadrature_error)
 R9_PREDICTIVE = perform(prior_predictive_and_posterior_predictive_checks)
 R10_VALIDATE = assess(calibration,sensitivity,known_posteriors,simulated_truth_and_reproducibility)
}

CONSTRAINTS {
 C1 = "Improper priors can yield improper posteriors."
 C2 = "Weak data can leave posterior results highly sensitive to prior assumptions."
 C3 = "MCMC diagnostics can look acceptable while important modes remain unexplored."
 C4 = "Variational approximations can underestimate posterior uncertainty."
 C5 = "Importance sampling can fail when proposal and posterior overlap is poor."
 C6 = "Marginal likelihood estimates can be unstable in high-dimensional models."
 C7 = "Hierarchical models can exhibit funnel geometry and strong posterior dependence."
 C8 = "Numerically unstable log-likelihood calculations can corrupt posterior inference."
 C9 = "Approximate Bayesian computation depends strongly on summary statistics, distance metrics and tolerance."
 C10 = "A concentrated posterior does not guarantee a well-specified model."
}

SEMANTIC_EQUIVALENCE {
 "BayesianComputation" <=> "BayesianInferenceComputation"
 "MaximumAPosteriori" <=> "MAP"
 "MarkovChainMonteCarlo" <=> "MCMC"
 "SequentialMonteCarlo" <=> "SMC"
 "VariationalInference" <=> "VI"
 "ApproximateBayesianComputation" <=> "ABC"
 "EffectiveSampleSize" <=> "ESS"
 "MonteCarloStandardError" <=> "MCSE"
}

VALIDATION {
 V1 = "Prior distributions and hyperparameters are documented."
 V2 = "Likelihood and data assumptions are documented."
 V3 = "Posterior-computation method and tuning are documented."
 V4 = "Parameter transformations and Jacobians are documented where used."
 V5 = "Sampling, optimisation or approximation convergence is assessed."
 V6 = "Posterior predictive checks are performed where appropriate."
 V7 = "Prior sensitivity is assessed when conclusions are prior-sensitive."
 V8 = "Known posterior, simulation-based calibration or benchmark tests are used where feasible."
 V9 = "Random seeds and parallel random-stream semantics are documented where stochastic computation is used."
 V10 = "Results are reproducible under documented model, data, seed, software and configuration settings."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 BAYESIAN_COMPUTATION_DEFINITION = 0.99
 PRIOR_LIKELIHOOD_REASONING = 0.99
 POSTERIOR_REASONING = 0.99
 MCMC_REASONING = 0.99
 VARIATIONAL_REASONING = 0.98
 MARGINAL_LIKELIHOOD_REASONING = 0.98
 POSTERIOR_PREDICTIVE_REASONING = 0.99
 POLICY = "Reduce confidence when prior, likelihood, parameterisation, approximation method, diagnostics, marginal-likelihood method or predictive checks are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,OntologyGrounding,BayesianComputationReasoning,
  PriorReasoning,LikelihoodReasoning,
  PosteriorReasoning,PosteriorPredictiveReasoning,
  MCMCReasoning,SMCReasoning,
  ImportanceSamplingReasoning,
  VariationalInferenceReasoning,
  LaplaceApproximationReasoning,
  ABCReasoning,MarginalLikelihoodReasoning,
  BayesFactorReasoning,HierarchicalModelReasoning,
  ParameterisationReasoning,
  ConvergenceDiagnosticReasoning,
  PosteriorCalibrationReasoning,
  ReproducibilityReasoning,
  KnowledgeGraphConstruction,RetrievalAugmentation,RecursiveInference
 ]
 NORMALISE {
  "Bayesian computation" -> BayesianComputation
  "Bayesian inference" -> BayesianComputation
  "MAP" -> MaximumAPosteriori
  "MCMC" -> MarkovChainMonteCarlo
  "SMC" -> SequentialMonteCarlo
  "VI" -> VariationalInference
  "ABC" -> ApproximateBayesianComputation
  "ESS" -> EffectiveSampleSize
  "MCSE" -> MonteCarloStandardError
 }
}

EXECUTION_RULES {
 E1 = "If priors and likelihoods are combined computationally to obtain posterior quantities, activate BayesianComputationOntology."
 E2 = "Define prior, likelihood and target posterior quantities before method selection."
 E3 = "Use stable log-density computation for numerically difficult models."
 E4 = "Choose exact, sampling-based, quadrature-based or approximate inference according to dimension and model structure."
 E5 = "Use parameter transformations and non-centred forms when geometry requires them."
 E6 = "Assess method-specific diagnostics before interpreting posterior summaries."
 E7 = "Perform prior and posterior predictive checks when possible."
 E8 = "Use simulation-based calibration or known-posterior tests for implementation validation where feasible."
 E9 = "Document seeds, algorithm settings, versions and provenance for reproducibility."
}

RECURSION {
 ENTRY = BayesianComputation
 EXPAND(BayesianComputation) -> [
  BayesianInference,BayesianStatistics,
  PosteriorComputation,BayesTheorem,
  PriorDistribution,PriorPredictiveDistribution,
  Likelihood,LogLikelihood,
  PosteriorDistribution,LogPosterior,
  Evidence,MarginalLikelihood,
  PosteriorPredictiveDistribution,
  PosteriorMean,PosteriorMedian,
  PosteriorMode,MaximumAPosteriori,MAP,
  CredibleInterval,HPD,
  ConjugatePrior,NonconjugateModel,
  HierarchicalModel,Hyperparameter,
  Hyperprior,PartialPooling,
  ConditionalIndependence,LatentVariable,
  DataAugmentation,MarkovChainMonteCarlo,
  MCMC,MetropolisHastings,
  GibbsSampling,HamiltonianMonteCarlo,
  HMC,NUTS,SliceSampling,
  SequentialMonteCarlo,SMC,
  ImportanceSampling,
  AnnealedImportanceSampling,
  BridgeSampling,ThermodynamicIntegration,
  VariationalInference,VI,
  MeanFieldVariationalInference,
  ELBO,LaplaceApproximation,
  ApproximateBayesianComputation,ABC,
  LikelihoodFreeInference,
  SimulationBasedInference,
  NumericalIntegration,Quadrature,
  MonteCarloIntegration,QuasiMonteCarlo,
  PosteriorApproximation,ApproximateInference,
  EffectiveSampleSize,ESS,
  MonteCarloStandardError,MCSE,
  RHat,DivergentTransition,
  Mixing,PosteriorCalibration,
  SimulationBasedCalibration,SBC,
  PriorSensitivity,WeaklyInformativePrior,
  ProperPrior,ImproperPrior,
  PriorPredictiveSimulation,
  PosteriorPredictiveCheck,PPC,
  BayesianModelComparison,BayesFactor,
  PosteriorModelProbability,
  BayesianModelAveraging,BMA,
  LOO,Marginalisation,
  NuisanceParameter,RaoBlackwellisation,
  Reparameterisation,CentredParameterisation,
  NonCentredParameterisation,
  WhiteningTransformation,
  JacobianAdjustment,
  CholeskyParameterisation,
  CovarianceMatrix,PrecisionMatrix,
  CholeskyDecomposition,SparseMatrix,
  BayesianRegression,BayesianGLM,
  GaussianProcessRelationship,
  StateSpaceModelRelationship,
  MixtureModel,RandomNumberGeneration,
  NumericalStability,FloatingPointArithmetic,
  LogSumExp,GradientOfLogPosterior,
  ComputationalComplexity,
  ParallelBayesianComputation,
  DistributedBayesianComputation,
  GPUBayesianComputation,
  Verification,KnownPosteriorTest,
  SimulationStudy,Benchmark,
  Reproducibility,Provenance
 ]
 EXPAND(BayesianComputationFamilies) -> [
  ConjugateAndAnalyticBayesianComputation,
  MCMCBasedPosteriorComputation,
  SequentialMonteCarloBayesianComputation,
  ImportanceAndEvidenceComputation,
  VariationalAndLaplaceApproximation,
  LikelihoodFreeAndSimulationBasedInference,
  HierarchicalBayesianComputation,
  ParallelDistributedAndGPUBayesianComputation
 ]
 STOP_CONDITION = "Stop when prior, likelihood, posterior target, approximation method, parameterisation, diagnostics, predictive checks, calibration and reproducibility are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Computational Statistics Ontology"
 RELATED_ONTOLOGIES = [
  "Computational Statistics Ontology",
  "Monte Carlo Method Ontology",
  "Markov Chain Monte Carlo Ontology",
  "Random Number Generation Ontology",
  "Statistical Sampling Ontology",
  "Probability and Statistics Ontologies",
  "Random Process Ontology",
  "Optimisation Computation",
  "Numerical Integration Ontology",
  "Numerical Linear Algebra Ontology 2",
  "Matrix Computation Ontology",
  "Numerical Error Ontology",
  "Numerical Stability Ontology 2",
  "Floating Point Arithmetic Ontology",
  "High Performance Computing Ontology 2",
  "Parallel Computing Ontology",
  "Distributed Computing Ontology",
  "GPU Computing Ontology"
 ]
 ROUTE_RULES = [
  "Bayesian Computation -> Computational Statistics Ontology for parent semantics",
  "Bayesian Computation -> Monte Carlo Method for posterior expectation estimation",
  "Bayesian Computation -> Markov Chain Monte Carlo for posterior sampling by Markov chains",
  "Bayesian Computation -> Random Number Generation for stochastic sampler reproducibility",
  "Bayesian Computation -> Statistical Sampling for distribution-draw semantics",
  "Bayesian Computation -> Probability and Statistics for prior, likelihood and posterior foundations",
  "Bayesian Computation -> Random Process for Markov and sequential stochastic methods",
  "Bayesian Computation -> Optimisation Computation for MAP and variational inference",
  "Bayesian Computation -> Numerical Integration for quadrature and marginal likelihood",
  "Bayesian Computation -> Numerical Linear Algebra and Matrix Computation for covariance and latent-variable computation",
  "Bayesian Computation -> Numerical Error and Stability for robust posterior evaluation",
  "Bayesian Computation -> Floating Point Arithmetic for log-density and reduction effects",
  "Bayesian Computation -> High Performance, Parallel, Distributed and GPU Computing for scalable inference"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible Bayesian algorithms, diagnostics or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve prior-versus-likelihood-versus-posterior distinction.",
  "Preserve posterior-density-versus-posterior-sample distinction.",
  "Preserve exact-versus-approximate inference distinction.",
  "Preserve posterior-inference-versus-predictive-inference distinction.",
  "Preserve posterior-probability-versus-model-evidence distinction.",
  "Preserve calibration-versus-convergence distinction.",
  "Preserve sampling-based-versus-optimisation-based approximation distinction.",
  "Preserve machine-readable Bayesian-computation semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Bayesian computation numerically evaluates or approximates posterior distributions and related Bayesian quantities."
 PRIOR_SUMMARY = "Prior distributions encode uncertainty before conditioning on observed data and should be checked through prior predictive implications."
 POSTERIOR_SUMMARY = "Likelihood and prior combine through Bayes' theorem to define the posterior distribution."
 METHOD_SUMMARY = "MCMC, SMC, importance sampling, quadrature, variational inference, Laplace approximation and ABC provide different posterior-computation strategies."
 PREDICTIVE_SUMMARY = "Posterior predictive distributions propagate posterior uncertainty into future or replicated observations."
 MODEL_COMPARISON_SUMMARY = "Marginal likelihoods, Bayes factors, predictive criteria and model averaging support Bayesian model comparison."
 VALIDATION_SUMMARY = "Bayesian computations should be checked through diagnostics, predictive checks, prior sensitivity, simulation-based calibration, known-posterior tests and reproducibility."
}

FLOW {
 INPUT
 -> detect_Bayesian_computation_context
 -> identify_data_model_parameters_latent_variables_prior_information_target_quantities_accuracy_goal_and_budget
 -> define_prior_hyperpriors_likelihood_conditional_independence_and_parameter_constraints
 -> compute_stable_unnormalised_log_posterior
 -> choose_conjugate_quadrature_MCMC_SMC_importance_variational_Laplace_ABC_or_hybrid_method
 -> choose_centred_noncentred_transformed_whitened_or_Cholesky_parameterisation
 -> configure_random_seed_streams_tuning_and_algorithm_settings
 -> execute_sampler_optimiser_quadrature_or_variational_approximation
 -> assess_ESS_MCSE_RHat_divergences_ELBO_importance_weights_or_quadrature_error
 -> compute_posterior_summaries_credible_intervals_and_target_expectations
 -> perform_prior_predictive_and_posterior_predictive_checks
 -> if_model_comparison_compute_predictive_or_marginal_likelihood_quantities_with_declared_method
 -> assess_prior_sensitivity_and_simulation_based_calibration_when_relevant
 -> if_large_parallelise_distribute_vectorise_or_GPU_accelerate_computation
 -> verify_against_conjugate_known_posterior_simulated_truth_or_reference_results
 -> document_model_seed_algorithm_versions_configuration_and_provenance
 -> route_to_related_computational_statistics_Monte_Carlo_MCMC_RNG_sampling_probability_random_process_optimisation_integration_linear_algebra_matrix_error_stability_floating_point_HPC_parallel_distributed_and_GPU_ontologies
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Bayesian Computation Ontology defines numerical evaluation and approximation of posterior distributions. It supports Bayes' theorem, prior, likelihood, posterior and marginal likelihood semantics, prior and posterior predictive distributions, posterior means, medians, modes and credible intervals, conjugate and hierarchical models, MCMC including Metropolis-Hastings, Gibbs, HMC and NUTS, sequential Monte Carlo, importance and rejection sampling, annealed importance sampling, bridge sampling and thermodynamic integration, variational inference and ELBO optimisation, Laplace and Gaussian approximation, approximate Bayesian computation and likelihood-free inference, numerical quadrature, effective sample size, Monte Carlo error, R-hat and divergence diagnostics, prior sensitivity, simulation-based calibration, Bayes factors, model averaging, centred and non-centred parameterisations, Jacobian transforms, Cholesky parameterisation, numerical stability, scalable parallel execution, verification and reproducibility."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-COMPUTATIONAL-STATISTICS-BAYESIAN-COMPUTATION-ONTOLOGY-V1"
}

}

END_FLOW