Monte Carlo Method Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Monte Carlo Method Ontology within the Kireth Research Semantic Library. Monte Carlo methods use repeated random sampling to estimate expectations, integrals, probabilities, distributions and uncertainty when analytical evaluation is difficult or impossible. This ontology supports random-number generation, sampling distributions, Monte Carlo estimators, convergence, central-limit behaviour, variance reduction, importance sampling, stratification, antithetic sampling, control variates, rare-event estimation, quasi-Monte Carlo relationships, parallel execution, diagnostics, numerical stability, verification and reproducibility.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = MonteCarloMethod
 ENTITY_CLASS = RandomSamplingEstimationFramework
 PRIMARY_CONTEXT = ComputationalStatistics
 DEFINITION = "A Monte Carlo method is a computational technique that uses repeated random sampling to estimate mathematical or statistical quantities."
 SYMBOLIC_FORMS = [
  "I = E[f(X)]",
  "I_hat_N = (1/N) * sum_(i=1)^N f(X_i)",
  "Var(I_hat_N)=Var(f(X))/N",
  "SE(I_hat_N)=sigma/sqrt(N)"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for random sampling, Monte Carlo estimation, sampling error, convergence, variance reduction, rare-event computation, diagnostics, parallelisation and reproducibility."
}

SCOPE {
 INCLUDE = [
  MonteCarloMethod,MonteCarlo,MonteCarloSimulation,
  MonteCarloEstimator,RandomSamplingMethod,
  SimulationBasedEstimation,
  ExpectationEstimation,IntegralEstimation,
  ProbabilityEstimation,DistributionEstimation,
  StochasticSimulation,
  RandomVariable,RandomVector,
  ProbabilityDistribution,TargetDistribution,
  SamplingDistribution,ProposalDistribution,
  Sample,RandomSample,IndependentSample,
  IndependentIdenticallyDistributedSample,IIDSample,
  SampleSize,NumberOfSamples,
  Replicate,Trial,SimulationRun,
  RandomNumberGeneration,RNG,
  PseudorandomNumberGenerator,PRNG,
  RandomSeed,RandomStream,
  IndependentRandomStream,
  CounterBasedRNG,
  QuasiRandomRelationship,
  DistributionSampling,
  UniformSampling,NormalSampling,
  ExponentialSampling,BernoulliSampling,
  CategoricalSampling,
  InverseTransformSampling,
  RejectionSampling,
  AcceptanceRejection,
  CompositionSampling,
  BoxMullerRelationship,
  TransformationMethod,
  MonteCarloIntegral,
  MultidimensionalIntegration,
  HighDimensionalIntegration,
  IndicatorFunction,
  EventProbability,
  RareEventProbability,
  FailureProbability,
  ReliabilityEstimation,
  TailProbability,
  MonteCarloMean,
  SampleMean,SampleVariance,
  UnbiasedEstimator,BiasedEstimator,
  ConsistentEstimator,
  MeanSquaredError,MSE,
  Bias,Variance,
  StandardError,MonteCarloStandardError,MCSE,
  ConfidenceInterval,
  CentralLimitTheorem,CLT,
  LawOfLargeNumbers,LLN,
  WeakLawOfLargeNumbers,
  StrongLawOfLargeNumbers,
  ConvergenceInProbability,
  AlmostSureConvergence,
  DistributionalConvergence,
  RootNConvergence,
  MonteCarloConvergence,
  ErrorRate,SamplingError,
  MonteCarloError,
  StatisticalError,
  RelativeError,AbsoluteError,
  CoefficientOfVariation,
  RelativeStandardError,
  VarianceReduction,
  ControlVariate,ControlVariates,
  AntitheticVariate,AntitheticSampling,
  StratifiedSampling,Stratum,
  ProportionalAllocation,
  OptimalAllocation,
  ImportanceSampling,
  ImportanceWeight,
  LikelihoodRatioWeight,
  SelfNormalisedImportanceSampling,
  WeightedEstimator,
  EffectiveSampleSize,ESS,
  SequentialImportanceSamplingRelationship,
  ConditionalMonteCarlo,
  RaoBlackwellisationRelationship,
  CommonRandomNumbers,
  CorrelatedSampling,
  LatinHypercubeSampling,LHS,
  OrthogonalSamplingRelationship,
  MultilevelMonteCarlo,MLMC,
  MultiIndexMonteCarloRelationship,
  NestedMonteCarlo,
  DoubleLoopMonteCarlo,
  SplittingMethod,
  SubsetSimulationRelationship,
  RareEventSimulation,
  ExponentialTilting,
  ChangeOfMeasure,
  ImportanceDistribution,
  AdaptiveImportanceSampling,
  CrossEntropyMethodRelationship,
  QuasiMonteCarlo,QMC,
  LowDiscrepancySequence,
  SobolSequence,HaltonSequence,
  ScrambledSequence,
  RandomisedQuasiMonteCarlo,RQMC,
  Discrepancy,StarDiscrepancy,
  NumericalIntegrationRelationship,
  BayesianComputationRelationship,
  MarkovChainMonteCarloRelationship,
  SequentialMonteCarloRelationship,
  BootstrapRelationship,
  StochasticOptimisationRelationship,
  UncertaintyQuantification,
  SensitivityAnalysis,
  SobolSensitivityRelationship,
  RiskEstimation,
  ValueAtRiskRelationship,
  ExpectedShortfallRelationship,
  ReliabilityAnalysis,
  OptionPricingRelationship,
  ComputationalPhysicsRelationship,
  ParticleSimulationRelationship,
  MolecularSimulationRelationship,
  FinancialSimulationRelationship,
  QueueingSimulationRelationship,
  RandomWalkSimulation,
  DiffusionSimulation,
  BrownianMotionSimulationRelationship,
  ParallelMonteCarlo,
  DistributedMonteCarlo,
  GPUMonteCarlo,
  VectorisedMonteCarlo,
  BatchMonteCarlo,
  EnsembleMonteCarlo,
  EmbarrassinglyParallel,
  ParallelRandomNumberGeneration,
  SeedSplitting,
  LeapfroggingRelationship,
  SequenceSplittingRelationship,
  StreamSplitting,
  Substream,
  ReproducibleRandomStream,
  LoadBalancing,
  DynamicLoadBalancing,
  WorkPartitioning,
  SamplePartitioning,
  Reduction,ParallelReduction,
  Summation,PairwiseSummation,
  CompensatedSummation,
  KahanSummationRelationship,
  FloatingPointArithmetic,
  RoundOffError,MachinePrecision,
  Overflow,Underflow,
  NumericalStability,
  ReproducibleSummation,
  DeterministicReduction,
  OrderDependentReduction,
  ComputationalComplexity,
  TimeComplexity,SpaceComplexity,
  CostPerSample,TotalCost,
  AccuracyCostTradeoff,
  VarianceCostProduct,
  EfficiencyMetric,
  FigureOfMerit,
  SequentialStoppingRule,
  FixedSampleStopping,
  PrecisionBasedStopping,
  ConfidenceIntervalStopping,
  AdaptiveSampleSize,
  PilotRun,
  BurnInNotRequiredForIIDMonteCarlo,
  BatchMeansRelationship,
  IndependentReplication,
  ConvergenceDiagnostic,
  RunningMean,
  RunningVariance,
  ErrorEstimate,
  ConfidenceSequenceRelationship,
  CoverageProbability,
  Calibration,
  Verification,CodeVerification,
  KnownAnswerTest,
  ReferenceIntegral,
  AnalyticalSolution,
  Benchmark,
  Validation,
  Reproducibility,Repeatability,
  Determinism,Nondeterminism,
  Provenance,Configuration,
  SoftwareVersion,LibraryVersion,
  RNGVersion,SeedMetadata
 ]
 EXCLUDE_AS_PRIMARY = [
  MarkovChainMonteCarloWhenDependenceStructureIsPrimary,
  DeterministicQuadratureWithoutRandomSampling,
  BootstrapWhenResamplingObservedDataIsPrimary,
  QuasiMonteCarloWhenDeterministicLowDiscrepancySequencesArePrimary
 ]
 DISAMBIGUATION_RULE = "When a quantity is estimated by averaging or aggregating outcomes from repeated random samples, interpret the context as a Monte Carlo method."
}

CORE_CONCEPTS {
 MonteCarloMethod = "Estimation through repeated random sampling."
 MonteCarloEstimator = "Statistic computed from simulated samples to approximate a target quantity."
 MonteCarloError = "Sampling uncertainty caused by finite Monte Carlo sample size."
 VarianceReduction = "Technique reducing estimator variance without proportionally increasing sample count."
 ImportanceSampling = "Sampling from a proposal distribution and reweighting to estimate a target quantity."
 ControlVariate = "Variance-reduction method using a correlated quantity with known expectation."
 AntitheticSampling = "Variance-reduction method using negatively correlated paired samples."
 StratifiedSampling = "Variance-reduction method sampling separately from defined subregions or strata."
}

RELATIONS {
 MonteCarloMethod --is_child_of--> ComputationalStatistics
 MonteCarloMethod --uses--> RandomSampling
 MonteCarloEstimator --approximates--> TargetQuantity
 SampleMean --is_type_of--> MonteCarloEstimator
 LawOfLargeNumbers --supports--> MonteCarloConsistency
 CentralLimitTheorem --supports--> MonteCarloStandardError
 VarianceReduction --reduces--> EstimatorVariance
 ImportanceSampling --uses--> ProposalDistribution
 ImportanceSampling --produces--> ImportanceWeights
 ControlVariate --uses--> KnownExpectation
 AntitheticSampling --uses--> NegativeCorrelation
 StratifiedSampling --partitions--> SamplingSpace
 MultilevelMonteCarlo --combines--> MultipleResolutionLevels
 ParallelMonteCarlo --distributes--> IndependentSamples
 Verification --checks--> MonteCarloImplementation
 Reproducibility --requires--> RNGAndSeedMetadata
}

OPERATORS {
 MC_ESTIMATE(f,X) := (1/N)*sum_i f(X_i)
 MC_VARIANCE(f,X) := sample_variance({f(X_i)})/N
 MC_STANDARD_ERROR(f,X) := sqrt(sample_variance({f(X_i)})/N)
 CONTROL_VARIATE_ESTIMATE(Y,C,beta,mu_C) := mean(Y-beta*(C-mu_C))
 ANTITHETIC_PAIR_ESTIMATE(f,U) := 0.5*(f(U)+f(1-U))
 IMPORTANCE_ESTIMATE(f,x,w) := sum_i w_i*f(x_i)/sum_i w_i
 ESS(weights) := 1/sum_i(normalised_weight_i^2)
 RELATIVE_ERROR(estimate,se) := se/abs(estimate)
}

AXIOMS {
 A1 = "Monte Carlo methods estimate target quantities through random samples."
 A2 = "For independent finite-variance sampling, standard Monte Carlo error generally decreases at order N^(-1/2)."
 A3 = "Increasing sample count reduces sampling error but not model error."
 A4 = "Variance reduction improves efficiency by lowering estimator variance for comparable cost."
 A5 = "Importance sampling requires adequate overlap between proposal and target distributions."
 A6 = "Parallel Monte Carlo requires valid random-stream independence or reproducible stream partitioning."
 A7 = "Finite-precision summation can affect large Monte Carlo reductions."
 A8 = "Monte Carlo results require explicit target, sampling, error, RNG and reproducibility semantics."
}

REASONING {
 R1_CONTEXT = identify(target_quantity,target_distribution,integrand,event,accuracy_goal,confidence_goal,budget)
 R2_SAMPLER = choose(direct,inverse_transform,rejection,importance,stratified_or_specialised_sampling)
 R3_RNG = configure(generator,seed,streams_and_parallel_substreams)
 R4_ESTIMATOR = define(sample_function,weights,normalisation_and_bias_properties)
 R5_VARIANCE = estimate_variance_and_select_control_antithetic_stratified_importance_or_multilevel_reduction_if_useful
 R6_EXECUTION = generate_samples_evaluate_model_and_accumulate_statistics
 R7_ERROR = compute(MCSE,relative_error,confidence_interval_and_effective_sample_size_if_weighted)
 R8_STOP = apply(fixed_sample_or_precision_based_stopping_rule)
 R9_SCALE = if_large distribute_samples_across_threads_processes_GPUs_or_batches
 R10_VALIDATE = compare(reference_integrals,analytical_results,replicated_runs_and_reproducibility)
}

CONSTRAINTS {
 C1 = "Monte Carlo convergence can be slow when very high precision is required."
 C2 = "Infinite or very large estimator variance invalidates standard error assumptions."
 C3 = "Rare events can be missed entirely under naive sampling."
 C4 = "Poor importance proposals can produce extreme weights and unstable estimates."
 C5 = "Variance-reduction methods can fail when their correlation or stratification assumptions are poor."
 C6 = "Random streams that overlap can bias parallel Monte Carlo results."
 C7 = "Stopping on an unstable error estimate can give misleading precision claims."
 C8 = "Model evaluation cost can dominate sampling overhead."
 C9 = "Floating-point summation error can accumulate over very large sample counts."
 C10 = "Low Monte Carlo error does not imply low model or parameter uncertainty."
}

SEMANTIC_EQUIVALENCE {
 "MonteCarloMethod" <=> "MonteCarlo"
 "MonteCarloStandardError" <=> "MCSE"
 "EffectiveSampleSize" <=> "ESS"
 "LatinHypercubeSampling" <=> "LHS"
 "MultilevelMonteCarlo" <=> "MLMC"
 "QuasiMonteCarlo" <=> "QMC"
 "RandomisedQuasiMonteCarlo" <=> "RQMC"
}

VALIDATION {
 V1 = "Target quantity and estimator are documented."
 V2 = "Sampling distribution and sampling algorithm are documented."
 V3 = "RNG, seed and random-stream semantics are documented."
 V4 = "Sample size and stopping rule are documented."
 V5 = "Monte Carlo standard error or equivalent uncertainty is reported."
 V6 = "Variance-reduction assumptions are documented where used."
 V7 = "Importance weights and effective sample size are checked where relevant."
 V8 = "Parallel random streams are checked for valid partitioning."
 V9 = "Known-answer, analytical or benchmark comparisons are used where feasible."
 V10 = "Results are reproducible under documented seed, RNG, software and reduction settings."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 MONTE_CARLO_DEFINITION = 0.99
 ERROR_REASONING = 0.99
 VARIANCE_REDUCTION_REASONING = 0.99
 IMPORTANCE_SAMPLING_REASONING = 0.99
 PARALLEL_MONTE_CARLO_REASONING = 0.98
 RARE_EVENT_REASONING = 0.98
 QMC_RELATIONSHIP_REASONING = 0.98
 POLICY = "Reduce confidence when target quantity, sampling distribution, RNG, sample size, variance, rare-event behaviour or stopping rule are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,OntologyGrounding,MonteCarloReasoning,
  RandomSamplingReasoning,EstimatorReasoning,
  MonteCarloErrorReasoning,VarianceReductionReasoning,
  ControlVariateReasoning,AntitheticSamplingReasoning,
  StratifiedSamplingReasoning,ImportanceSamplingReasoning,
  RareEventReasoning,MultilevelMonteCarloReasoning,
  QuasiMonteCarloRelationshipReasoning,
  ParallelMonteCarloReasoning,RNGReasoning,
  StoppingRuleReasoning,ReproducibilityReasoning,
  VerificationReasoning,KnowledgeGraphConstruction,
  RetrievalAugmentation,RecursiveInference
 ]
 NORMALISE {
  "Monte Carlo method" -> MonteCarloMethod
  "Monte Carlo" -> MonteCarloMethod
  "MCSE" -> MonteCarloStandardError
  "ESS" -> EffectiveSampleSize
  "LHS" -> LatinHypercubeSampling
  "MLMC" -> MultilevelMonteCarlo
  "QMC" -> QuasiMonteCarlo
  "RQMC" -> RandomisedQuasiMonteCarlo
 }
}

EXECUTION_RULES {
 E1 = "If a quantity is estimated from repeated random samples, activate MonteCarloMethodOntology."
 E2 = "Define target quantity, sampling distribution and estimator before simulation."
 E3 = "Document RNG, seed and random-stream semantics."
 E4 = "Estimate Monte Carlo error rather than reporting only a point estimate."
 E5 = "Use variance reduction when simulation cost or estimator variance makes naive Monte Carlo inefficient."
 E6 = "For importance sampling, monitor weight concentration and effective sample size."
 E7 = "For rare events, use specialised sampling when naive occurrence counts are insufficient."
 E8 = "For parallel execution, use independent or formally partitioned random streams."
 E9 = "Validate against known results, repeated runs and reproducibility tests."
}

RECURSION {
 ENTRY = MonteCarloMethod
 EXPAND(MonteCarloMethod) -> [
  MonteCarlo,MonteCarloSimulation,
  MonteCarloEstimator,RandomSamplingMethod,
  ExpectationEstimation,IntegralEstimation,
  ProbabilityEstimation,RandomVariable,
  ProbabilityDistribution,RandomSample,
  IIDSample,SampleSize,Replicate,
  RandomNumberGeneration,PRNG,
  RandomSeed,RandomStream,
  CounterBasedRNG,DistributionSampling,
  UniformSampling,NormalSampling,
  InverseTransformSampling,RejectionSampling,
  MonteCarloIntegral,IndicatorFunction,
  RareEventProbability,SampleMean,
  SampleVariance,UnbiasedEstimator,
  ConsistentEstimator,Bias,Variance,
  MeanSquaredError,MonteCarloStandardError,
  MCSE,ConfidenceInterval,
  CentralLimitTheorem,LawOfLargeNumbers,
  RootNConvergence,MonteCarloError,
  RelativeError,VarianceReduction,
  ControlVariate,AntitheticSampling,
  StratifiedSampling,ImportanceSampling,
  ImportanceWeight,SelfNormalisedImportanceSampling,
  EffectiveSampleSize,ESS,
  ConditionalMonteCarlo,CommonRandomNumbers,
  LatinHypercubeSampling,LHS,
  MultilevelMonteCarlo,MLMC,
  NestedMonteCarlo,RareEventSimulation,
  ChangeOfMeasure,AdaptiveImportanceSampling,
  QuasiMonteCarlo,QMC,
  SobolSequence,HaltonSequence,
  RandomisedQuasiMonteCarlo,RQMC,
  UncertaintyQuantification,
  SensitivityAnalysis,RiskEstimation,
  ReliabilityAnalysis,RandomWalkSimulation,
  ParallelMonteCarlo,DistributedMonteCarlo,
  GPUMonteCarlo,VectorisedMonteCarlo,
  EnsembleMonteCarlo,EmbarrassinglyParallel,
  ParallelRandomNumberGeneration,
  SeedSplitting,StreamSplitting,
  LoadBalancing,ParallelReduction,
  CompensatedSummation,
  FloatingPointArithmetic,RoundOffError,
  NumericalStability,ComputationalComplexity,
  CostPerSample,SequentialStoppingRule,
  PrecisionBasedStopping,PilotRun,
  RunningMean,ErrorEstimate,
  Verification,KnownAnswerTest,
  Benchmark,Reproducibility,Provenance
 ]
 EXPAND(MonteCarloFamilies) -> [
  CrudeMonteCarlo,
  VarianceReducedMonteCarlo,
  ImportanceSamplingMonteCarlo,
  RareEventMonteCarlo,
  MultilevelMonteCarlo,
  QuasiMonteCarloRelationships,
  ParallelAndDistributedMonteCarlo,
  MonteCarloUncertaintyEstimation
 ]
 STOP_CONDITION = "Stop when target, sampling, RNG, estimator, variance, error, stopping, parallelism and validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Computational Statistics Ontology"
 RELATED_ONTOLOGIES = [
  "Computational Statistics Ontology",
  "Computational Statistics",
  "Probability and Statistics Ontologies",
  "Random Process Ontology",
  "Markov Chain Monte Carlo Ontology",
  "Sequential Monte Carlo Ontology",
  "Bootstrap Ontology",
  "Numerical Integration Ontology",
  "Quasi Monte Carlo Ontology",
  "Uncertainty Quantification Ontologies",
  "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 = [
  "Monte Carlo Method -> Computational Statistics Ontology for parent semantics",
  "Monte Carlo Method -> Probability and Statistics for expectation and variance foundations",
  "Monte Carlo Method -> Random Process for stochastic sampling relationships",
  "Monte Carlo Method -> Markov Chain Monte Carlo when dependent Markov-chain samples are primary",
  "Monte Carlo Method -> Sequential Monte Carlo for evolving weighted-particle populations",
  "Monte Carlo Method -> Bootstrap when resampling observed data is primary",
  "Monte Carlo Method -> Numerical Integration for integral-estimation semantics",
  "Monte Carlo Method -> Quasi Monte Carlo for low-discrepancy deterministic or randomised sequences",
  "Monte Carlo Method -> Uncertainty Quantification for propagation and risk estimation",
  "Monte Carlo Method -> Numerical Error and Stability for estimator and accumulation errors",
  "Monte Carlo Method -> Floating Point Arithmetic for large reductions",
  "Monte Carlo Method -> High Performance, Parallel, Distributed and GPU Computing for scalable sampling"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible Monte Carlo estimators, variance-reduction methods or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve random-sampling-versus-deterministic-quadrature distinction.",
  "Preserve sampling-error-versus-model-error distinction.",
  "Preserve estimator-bias-versus-estimator-variance distinction.",
  "Preserve crude-versus-variance-reduced Monte Carlo distinction.",
  "Preserve direct-sampling-versus-importance-sampling distinction.",
  "Preserve independent-sampling-versus-Markov-dependent sampling distinction.",
  "Preserve point-estimate-versus-Monte-Carlo-error distinction.",
  "Preserve machine-readable Monte-Carlo semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "A Monte Carlo method estimates mathematical or statistical quantities through repeated random sampling."
 ERROR_SUMMARY = "For standard independent finite-variance Monte Carlo, sampling error decreases proportionally to the inverse square root of sample size."
 VARIANCE_SUMMARY = "Control variates, antithetic samples, stratification and importance sampling can reduce estimator variance."
 RARE_EVENT_SUMMARY = "Rare-event probabilities often require importance sampling, splitting or related specialised techniques."
 PARALLEL_SUMMARY = "Independent sampling makes many Monte Carlo workloads naturally parallel, provided random streams are validly partitioned."
 STOPPING_SUMMARY = "Sample size can be fixed in advance or adapted using Monte Carlo standard error and precision targets."
 VALIDATION_SUMMARY = "Monte Carlo estimates should document RNG, sample count, uncertainty, variance-reduction assumptions, reference checks and reproducibility."
}

FLOW {
 INPUT
 -> detect_Monte_Carlo_method_context
 -> identify_target_quantity_distribution_integrand_event_accuracy_goal_confidence_goal_and_budget
 -> choose_direct_inverse_transform_rejection_importance_stratified_multilevel_or_specialised_sampler
 -> configure_PRNG_seed_streams_and_parallel_substreams
 -> define_estimator_weights_normalisation_bias_and_target_expectation
 -> estimate_pilot_variance_if_needed
 -> choose_control_variate_antithetic_stratified_importance_multilevel_or_other_variance_reduction_if_beneficial
 -> generate_samples_and_evaluate_model
 -> accumulate_estimate_variance_MCSE_relative_error_and_confidence_interval
 -> if_weighted_compute_weight_concentration_and_effective_sample_size
 -> apply_fixed_sample_or_precision_based_stopping_rule
 -> if_large_parallelise_samples_across_threads_processes_GPUs_or_batches
 -> verify_against_analytical_reference_known_answer_or_replicated_results
 -> assess_random_stream_validity_numerical_accumulation_and_reproducibility
 -> document_RNG_seed_sample_count_software_versions_reduction_method_and_provenance
 -> route_to_related_computational_statistics_probability_random_process_MCMC_SMC_bootstrap_numerical_integration_QMC_uncertainty_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 Monte Carlo Method Ontology defines estimation through repeated random sampling. It supports direct distribution sampling, inverse-transform and rejection methods, expectation, integral and probability estimation, sample means and variances, law-of-large-numbers and central-limit reasoning, Monte Carlo standard errors and confidence intervals, variance reduction through control variates, antithetic sampling and stratification, importance sampling and effective sample size, Latin hypercube, conditional and multilevel Monte Carlo, rare-event estimation and change of measure, quasi-Monte-Carlo relationships, risk and reliability estimation, parallel, distributed, vectorised and GPU sampling, random-stream partitioning, stable accumulation, fixed and adaptive stopping rules, verification against analytical or reference results, and reproducible seed, RNG and configuration semantics."
}

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

}

END_FLOW