Monte Carlo Simulation Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Monte Carlo Simulation Ontology within the Kireth Research Semantic Library. Monte Carlo simulation uses repeated random sampling to approximate expectations, probabilities, integrals, distributions, uncertainty and complex system behaviour. It connects applied statistical mathematics with probability, numerical integration, uncertainty quantification, Bayesian computation and computational mathematics.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Monte Carlo Simulation Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Applied Statistical Mathematics / Monte Carlo Simulation"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/applied-statistical-mathematics/monte-carlo-simulation-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = MonteCarloSimulation
 ENTITY_CLASS = RandomSamplingComputationalInferenceFramework
 PRIMARY_CONTEXT = AppliedStatisticalMathematics
 DEFINITION = "Monte Carlo simulation is the mathematical and computational use of repeated sampling to approximate quantities or system behaviour that are difficult to evaluate analytically."
 SYMBOLIC_FORMS = [
  "I=E[f(X)]",
  "I_hat=(1/N)sum_i f(X_i)",
  "SE(I_hat) approximately sigma/sqrt(N)",
  "P(A) approximately (1/N)sum_i 1_A(X_i)"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for random sampling, simulation estimators, numerical integration, uncertainty propagation, variance reduction, convergence and reproducibility."
}

SCOPE {
 INCLUDE = [
  MonteCarloSimulation, MonteCarloMethod, RandomSampling, IIDSample,
  PseudoRandomNumberGenerator, RandomSeed, RandomVariable,
  ProbabilityDistribution, InverseTransformSampling, RejectionSampling,
  ImportanceSampling, ProposalDistribution, ImportanceWeight,
  StratifiedSampling, LatinHypercubeSampling, AntitheticVariates,
  ControlVariates, CommonRandomNumbers, ConditionalMonteCarlo,
  VarianceReduction, MonteCarloEstimator, SampleMeanEstimator,
  SampleVariance, StandardError, MonteCarloError, Bias, Variance,
  MeanSquaredError, Consistency, LawOfLargeNumbers, CentralLimitTheorem,
  ConfidenceInterval, NumericalIntegration, HighDimensionalIntegration,
  ExpectationEstimation, ProbabilityEstimation, QuantileEstimation,
  TailProbability, RareEventSimulation, SplittingMethod,
  SequentialMonteCarlo, QuasiMonteCarlo, LowDiscrepancySequence,
  SobolSequence, HaltonSequence, RandomisedQuasiMonteCarlo,
  UncertaintyPropagation, InputUncertainty, OutputUncertainty,
  ParameterUncertainty, ModelUncertainty, SensitivityAnalysis,
  SobolSensitivityIndex, RiskSimulation, ReliabilitySimulation,
  FailureProbability, NestedMonteCarlo, MultilevelMonteCarlo,
  MarkovChainMonteCarlo, EffectiveSampleSize, ParallelMonteCarlo,
  DistributedSimulation, Reproducibility, ConvergenceDiagnostic,
  StoppingRule
 ]
 EXCLUDE_AS_PRIMARY = [
  DeterministicSimulationWithoutRandomSampling,
  ClosedFormProbabilityCalculationWithoutSimulation,
  MCMCAsPrimaryPosteriorSamplingFramework
 ]
 DISAMBIGUATION_RULE = "When repeated random or low-discrepancy sampling approximates expectations, probabilities, integrals, distributions or uncertainty, interpret the context as Monte Carlo simulation."
}

CORE_CONCEPTS {
 MonteCarloSimulation = "Approximation of mathematical quantities or system behaviour through repeated sampling."
 MonteCarloEstimator = "Statistic computed from simulated samples to estimate a target quantity."
 MonteCarloError = "Sampling error caused by using finitely many simulation draws."
 VarianceReduction = "Method reducing estimator variance for a fixed computational budget."
 ImportanceSampling = "Sampling from a proposal distribution and reweighting observations toward a target distribution."
 QuasiMonteCarlo = "Numerical integration using low-discrepancy point sequences."
 RareEventSimulation = "Methods designed to estimate very small probabilities efficiently."
 UncertaintyPropagation = "Propagation of uncertain model inputs into output uncertainty."
}

RELATIONS {
 MonteCarloSimulation --is_child_of--> AppliedStatisticalMathematics
 MonteCarloSimulation --uses--> RandomSampling
 RandomSampling --draws_from--> ProbabilityDistribution
 MonteCarloEstimator --approximates--> TargetQuantity
 LawOfLargeNumbers --supports--> MonteCarloConsistency
 CentralLimitTheorem --supports--> MonteCarloStandardError
 MonteCarloError --decreases_with--> SampleSize
 VarianceReduction --improves--> EstimatorEfficiency
 ImportanceSampling --uses--> ProposalDistribution
 ImportanceWeight --corrects_for--> ProposalDistribution
 ControlVariates --reduce--> Variance
 StratifiedSampling --partitions--> SampleSpace
 QuasiMonteCarlo --uses--> LowDiscrepancySequence
 RareEventSimulation --estimates--> TailProbability
 UncertaintyPropagation --maps--> InputUncertaintyToOutputUncertainty
 MultilevelMonteCarlo --combines--> MultipleResolutionLevels
 ParallelMonteCarlo --distributes--> SimulationReplicates
}

OPERATORS {
 MONTE_CARLO_MEAN(f,X,N) := I_hat=(1/N)*sum_i=1^N f(X_i)
 MONTE_CARLO_VARIANCE(Y,N) := s2=(1/(N-1))*sum_i(Y_i-Y_bar)^2
 MONTE_CARLO_STANDARD_ERROR(s,N) := SE=s/sqrt(N)
 EVENT_PROBABILITY(A,X,N) := p_hat=(1/N)*sum_i 1_A(X_i)
 IMPORTANCE_ESTIMATE(f,p,q,X,N) := I_hat=(1/N)*sum_i f(X_i)*p(X_i)/q(X_i)
 SELF_NORMALISED_IMPORTANCE(f,w,X) := I_hat=sum_i w_i*f(X_i)/sum_i w_i
 CONTROL_VARIATE(Y,C,b,mu_C) := Y_cv=Y-b*(C-mu_C)
 ANTITHETIC_PAIR(U) := (U,1-U)
 EFFECTIVE_SAMPLE_SIZE(w) := ESS=(sum_i w_i)^2/sum_i w_i^2
 FAILURE_PROBABILITY(g,X,N) := p_f_hat=(1/N)*sum_i 1[g(X_i)<=0]
}

AXIOMS {
 A1 = "Monte Carlo estimation requires an explicit target quantity and sampling mechanism."
 A2 = "Finite Monte Carlo estimates contain simulation error."
 A3 = "Ordinary independent Monte Carlo commonly converges at standard-error order N^(-1/2) under suitable finite-variance conditions."
 A4 = "Variance-reduction methods must preserve or explicitly characterise estimator bias."
 A5 = "Importance sampling requires proposal support wherever the target contributes."
 A6 = "Random seeds support reproducibility but do not replace statistical replication."
 A7 = "Rare-event estimation requires adequate effective sampling of the relevant tail."
 A8 = "Monte Carlo conclusions require convergence and uncertainty validation."
}

REASONING {
 R1_CONTEXT = identify(target_quantity,input_distributions,model,accuracy_goal,compute_budget)
 R2_SAMPLING = choose(direct,rejection,importance,stratified,LHS,QMC_or_specialised_method)
 R3_GENERATION = generate(samples,record_seed_and_generator)
 R4_EVALUATION = evaluate(model_or_integrand)
 R5_ESTIMATION = compute(expectation,probability,quantile,integral_or_distribution)
 R6_ERROR = estimate(variance,standard_error,confidence_interval,bias_if_present)
 R7_EFFICIENCY = assess(variance_reduction,ESS,parallelism,cost)
 R8_RARE_EVENT = if_required select(importance_or_splitting_method)
 R9_SENSITIVITY = if_required compute(global_sensitivity_indices)
 R10_VALIDATION = check(convergence,replication,random_streams,stopping_rule)
}

CONSTRAINTS {
 C1 = "Increasing sample size does not correct model misspecification."
 C2 = "Ordinary Monte Carlo can be inefficient for rare events."
 C3 = "Extreme importance weights can make estimates unstable."
 C4 = "A poor proposal distribution can make importance sampling inefficient."
 C5 = "Pseudo-random samples depend on generator quality."
 C6 = "Quasi-Monte Carlo error analysis differs from IID Monte Carlo."
 C7 = "Nested Monte Carlo can have high computational cost."
 C8 = "Parallel simulation must avoid unintended random-stream overlap."
 C9 = "Visual stability alone is not a sufficient stopping criterion."
 C10 = "Simulation error must be distinguished from model and input uncertainty."
}

SEMANTIC_EQUIVALENCE {
 "MonteCarloSimulation" <=> "MonteCarloMethod"
 "PseudoRandomNumberGenerator" <=> "PRNG"
 "LatinHypercubeSampling" <=> "LHS"
 "QuasiMonteCarlo" <=> "QMC"
 "MultilevelMonteCarlo" <=> "MLMC"
 "MarkovChainMonteCarlo" <=> "MCMC"
}

VALIDATION {
 V1 = "Target quantity is explicit."
 V2 = "Input probability distributions are documented."
 V3 = "Random-number generator and seed strategy are documented where relevant."
 V4 = "Monte Carlo standard error or equivalent uncertainty is reported."
 V5 = "Convergence is assessed as sample size increases."
 V6 = "Importance-weight stability is checked where applicable."
 V7 = "Variance reduction is compared against a baseline."
 V8 = "Rare-event methods receive specialised validation."
 V9 = "Parallel random streams are appropriately constructed."
 V10 = "Simulation error is distinguished from parameter, input and model uncertainty."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 MONTE_CARLO_DEFINITION = 0.99
 SAMPLING_REASONING = 0.99
 ESTIMATOR_REASONING = 0.99
 ERROR_REASONING = 0.99
 IMPORTANCE_SAMPLING_REASONING = 0.98
 VARIANCE_REDUCTION_REASONING = 0.98
 QMC_REASONING = 0.98
 RARE_EVENT_REASONING = 0.97
 POLICY = "Reduce confidence when target quantity, input distributions, sample size, proposal distribution, dependence, random streams or stopping criteria are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch, OntologyGrounding, MonteCarloSimulationReasoning,
  RandomSamplingReasoning, NumericalIntegrationReasoning,
  ExpectationEstimationReasoning, ProbabilityEstimationReasoning,
  MonteCarloErrorReasoning, VarianceReductionReasoning,
  ImportanceSamplingReasoning, StratifiedSamplingReasoning,
  QuasiMonteCarloReasoning, RareEventReasoning,
  UncertaintyPropagationReasoning, SensitivityAnalysisReasoning,
  ReliabilitySimulationReasoning, MultilevelMonteCarloReasoning,
  ParallelSimulationReasoning, KnowledgeGraphConstruction,
  RetrievalAugmentation, RecursiveInference
 ]
 NORMALISE {
  "Monte Carlo simulation" -> MonteCarloSimulation
  "Monte Carlo method" -> MonteCarloSimulation
  "PRNG" -> PseudoRandomNumberGenerator
  "LHS" -> LatinHypercubeSampling
  "QMC" -> QuasiMonteCarlo
  "MLMC" -> MultilevelMonteCarlo
  "MCMC" -> MarkovChainMonteCarlo
 }
}

EXECUTION_RULES {
 E1 = "If repeated sampling approximates a mathematical or statistical target, activate MonteCarloSimulationOntology."
 E2 = "If direct sampling is feasible, use direct Monte Carlo as a baseline."
 E3 = "If variance is high, consider control variates, antithetic variates, stratification or importance sampling."
 E4 = "If integration is high-dimensional, consider Monte Carlo or quasi-Monte Carlo."
 E5 = "If rare events dominate, use specialised importance or splitting methods."
 E6 = "If uncertain inputs pass through a model, use uncertainty-propagation reasoning."
 E7 = "If multiple numerical resolutions exist, consider multilevel Monte Carlo."
 E8 = "If simulations are parallelised, validate random streams."
 E9 = "Any Monte Carlo estimate should include simulation-error and convergence assessment."
}

RECURSION {
 ENTRY = MonteCarloSimulation
 EXPAND(MonteCarloSimulation) -> [
  RandomSampling, IIDSample, PseudoRandomNumberGenerator, RandomSeed,
  RandomVariable, ProbabilityDistribution, RejectionSampling,
  ImportanceSampling, ProposalDistribution, ImportanceWeight,
  StratifiedSampling, LatinHypercubeSampling, AntitheticVariates,
  ControlVariates, VarianceReduction, MonteCarloEstimator,
  StandardError, MonteCarloError, Bias, Variance, LawOfLargeNumbers,
  CentralLimitTheorem, NumericalIntegration, ExpectationEstimation,
  ProbabilityEstimation, QuantileEstimation, RareEventSimulation,
  SequentialMonteCarlo, QuasiMonteCarlo, LowDiscrepancySequence,
  SobolSequence, HaltonSequence, UncertaintyPropagation,
  SensitivityAnalysis, SobolSensitivityIndex, RiskSimulation,
  ReliabilitySimulation, FailureProbability, NestedMonteCarlo,
  MultilevelMonteCarlo, MarkovChainMonteCarlo, EffectiveSampleSize,
  ParallelMonteCarlo, Reproducibility, ConvergenceDiagnostic, StoppingRule
 ]
 STOP_CONDITION = "Stop when target, sampling distribution, estimator, simulation error, efficiency, convergence and reproducibility are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Applied Statistical Mathematics"
 RELATED_ONTOLOGIES = [
  "Applied Statistical Mathematics",
  "Probability & Statistics Ontologies",
  "Bayesian Statistics Ontology",
  "Statistical Modelling Ontology",
  "Numerical Integration Ontologies",
  "Markov Chain Ontologies",
  "Uncertainty Quantification Ontologies",
  "Risk Mathematics Ontologies",
  "Computational Statistics Ontologies",
  "High Performance Computing Ontologies"
 ]
 ROUTE_RULES = [
  "Monte Carlo Simulation -> Applied Statistical Mathematics for parent semantics",
  "Monte Carlo Simulation -> Probability & Statistics for sampling foundations",
  "Monte Carlo Simulation -> Bayesian Statistics for posterior simulation",
  "Monte Carlo Simulation -> Statistical Modelling for model-based simulation",
  "Monte Carlo Simulation -> Numerical Integration for integral estimation",
  "Monte Carlo Simulation -> Markov Chains for MCMC-related sampling",
  "Monte Carlo Simulation -> Uncertainty Quantification for uncertainty propagation",
  "Monte Carlo Simulation -> Risk Mathematics for tail probabilities",
  "Monte Carlo Simulation -> Computational Statistics and HPC for scalable simulation"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible sampling, variance-reduction or simulation pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve repeated-sampling semantics.",
  "Preserve target-versus-proposal distinction.",
  "Preserve estimator-versus-simulation-error distinction.",
  "Preserve ordinary-versus-quasi-Monte-Carlo distinction.",
  "Preserve direct-versus-importance-sampling distinction.",
  "Preserve reproducibility semantics.",
  "Preserve simulation-versus-model uncertainty distinction.",
  "Preserve machine-readable Monte-Carlo semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Monte Carlo simulation approximates expectations, probabilities, distributions and system behaviour through repeated sampling."
 ESTIMATOR_SUMMARY = "Monte Carlo estimators average simulated outcomes to approximate target quantities."
 ERROR_SUMMARY = "Finite simulation produces Monte Carlo error that should be quantified."
 VARIANCE_SUMMARY = "Variance-reduction methods improve efficiency for a fixed simulation budget."
 QMC_SUMMARY = "Quasi-Monte Carlo uses low-discrepancy sequences for numerical integration."
 RARE_EVENT_SUMMARY = "Rare-event methods target regions that ordinary simulation may visit too infrequently."
 UNCERTAINTY_SUMMARY = "Monte Carlo methods propagate uncertain inputs through mathematical and computational models."
 VALIDATION_SUMMARY = "Monte Carlo results should be checked for convergence, sampling validity, simulation error and reproducibility."
}

FLOW {
 INPUT
 -> detect_Monte_Carlo_simulation_context
 -> identify_target_quantity_input_distributions_model_accuracy_goal_and_compute_budget
 -> select_sampling_method
 -> generate_samples_with_documented_random_stream
 -> evaluate_model_or_integrand
 -> compute_target_estimator
 -> estimate_variance_standard_error_and_confidence
 -> apply_variance_reduction_if_required
 -> assess_rare_events_sensitivity_or_multilevel_structure_if_required
 -> validate_convergence_random_stream_integrity_and_reproducibility
 -> route_to_applied_statistical_mathematics_probability_statistics_Bayesian_statistics_statistical_modelling_numerical_integration_uncertainty_quantification_risk_mathematics_computational_statistics_or_HPC_ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Monte Carlo Simulation Ontology defines computational inference through repeated sampling. It supports random and pseudo-random sampling, seeds and generators, probability distributions, rejection and importance sampling, stratification, Latin hypercube sampling, antithetic and control variates, variance reduction, Monte Carlo estimators, standard errors and convergence, numerical integration and probability estimation, rare-event simulation, sequential and quasi-Monte Carlo, Sobol and Halton sequences, uncertainty propagation, sensitivity analysis, risk and reliability simulation, nested and multilevel Monte Carlo, MCMC relationships, effective sample size, parallel simulation, stopping rules and reproducibility."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-APPLIED-MATHEMATICS-MONTE-CARLO-SIMULATION-ONTOLOGY-V1"
}

}

END_FLOW