Computational Statistics

AI INTERPRETATION NOTE
This Flow Algebra defines the Computational Statistics parent ontology within the Kireth Research Semantic Library. Computational statistics is the use of numerical, algorithmic and simulation-based methods to perform statistical estimation, inference, model fitting, uncertainty quantification and data analysis when analytical solutions are unavailable, impractical or computationally expensive. This ontology supports Monte Carlo methods, resampling, Bayesian computation, optimisation, likelihood methods, numerical integration, stochastic simulation, bootstrap and permutation methods, MCMC, importance sampling, sequential Monte Carlo, EM algorithms, approximate inference, high-dimensional statistics, computational complexity, numerical stability, reproducibility and scalable statistical computation.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = ComputationalStatistics
 ENTITY_CLASS = StatisticalComputationFramework
 PRIMARY_CONTEXT = ComputationalMathematicsOntologies
 DEFINITION = "Computational statistics is the use of algorithms, numerical methods and simulation to perform statistical estimation, inference, model fitting, prediction and uncertainty analysis."
 SYMBOLIC_FORMS = [
  "theta_hat = argmax_theta L(theta|data)",
  "E[f(X)] approximately (1/N) * sum_i f(X_i)",
  "p(theta|data) proportional to p(data|theta)*p(theta)",
  "bootstrap_statistic = T(data*)"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for simulation-based inference, resampling, numerical estimation, Bayesian computation, optimisation, likelihood evaluation, uncertainty and reproducibility."
}

SCOPE {
 INCLUDE = [
  ComputationalStatistics,StatisticalComputing,
  NumericalStatistics,SimulationBasedStatistics,
  StatisticalInference,StatisticalEstimation,
  StatisticalModel,ProbabilisticModel,
  Parameter,Estimator,Estimate,
  Statistic,SufficientStatistic,
  Likelihood,LogLikelihood,
  MaximumLikelihood,MLE,
  PenalisedLikelihood,ProfileLikelihood,
  BayesianInference,BayesianComputation,
  PriorDistribution,PosteriorDistribution,
  PosteriorPredictiveDistribution,
  MarginalLikelihood,Evidence,
  BayesFactorRelationship,
  MonteCarlo,MonteCarloSimulation,
  MonteCarloEstimator,LawOfLargeNumbers,
  CentralLimitTheorem,MonteCarloError,
  VarianceReduction,ControlVariate,
  AntitheticVariate,StratifiedSampling,
  ImportanceSampling,ImportanceWeight,
  SelfNormalisedImportanceSampling,
  RejectionSampling,AcceptanceRejection,
  MarkovChainMonteCarlo,MCMC,
  MetropolisAlgorithm,MetropolisHastings,
  GibbsSampling,BlockedGibbs,
  HamiltonianMonteCarlo,HMC,
  NoUTurnSampler,NUTS,
  SliceSampling,AdaptiveMCMC,
  RandomWalkMetropolis,
  LangevinMonteCarlo,
  MetropolisAdjustedLangevinAlgorithm,MALA,
  SequentialMonteCarlo,SMC,
  ParticleFilter,ParticleSmoother,
  Resampling,MultinomialResampling,
  StratifiedResampling,SystematicResampling,
  EffectiveSampleSize,ESS,
  AnnealedImportanceSampling,AIS,
  SequentialImportanceSampling,
  Bootstrap,NonparametricBootstrap,
  ParametricBootstrap,
  ResidualBootstrap,BlockBootstrap,
  MovingBlockBootstrap,StationaryBootstrap,
  BootstrapConfidenceInterval,
  PercentileInterval,BCaInterval,
  PermutationTest,RandomisationTest,
  ExactTestRelationship,
  Jackknife,DeleteOneJackknife,
  CrossValidation,LeaveOneOut,
  KFoldCrossValidation,
  NestedCrossValidation,
  ModelSelection,ModelComparison,
  InformationCriterion,AIC,BIC,
  WAICRelationship,LOORelationship,
  PredictiveValidation,
  ExpectationMaximisation,EM,
  GeneralisedEM,GEM,
  VariationalInference,VI,
  MeanFieldVariationalInference,
  EvidenceLowerBound,ELBO,
  LaplaceApproximation,
  ExpectationPropagationRelationship,
  ApproximateBayesianComputation,ABC,
  SyntheticLikelihood,
  LikelihoodFreeInference,
  SimulationBasedInference,
  NumericalIntegration,Quadrature,
  GaussianQuadrature,AdaptiveQuadrature,
  MonteCarloIntegration,
  QuasiMonteCarlo,QMC,
  LowDiscrepancySequence,
  SobolSequence,HaltonSequence,
  NumericalOptimisation,
  GradientDescent,NewtonMethod,
  QuasiNewtonMethod,BFGS,LBFGS,
  ConjugateGradient,
  TrustRegion,LineSearch,
  StochasticOptimisation,
  StochasticGradientDescent,
  CoordinateDescent,
  ProximalGradient,
  ADMMRelationship,
  RootFinding,NewtonRaphson,
  ScoreEquation,
  FisherScoring,
  IterativelyReweightedLeastSquares,IRLS,
  LinearRegression,GeneralisedLinearModel,GLM,
  LogisticRegression,PoissonRegression,
  ProbitModel,
  MixedModel,HierarchicalModel,
  GeneralisedLinearMixedModel,GLMM,
  GaussianProcess,GP,
  StateSpaceModel,HiddenMarkovModel,HMM,
  TimeSeriesModel,
  ARIMA,VARRelationship,
  SurvivalModel,CoxModel,
  CensoredData,
  LatentVariableModel,
  MixtureModel,GaussianMixtureModel,GMM,
  FactorAnalysis,
  PrincipalComponentAnalysis,PCA,
  IndependentComponentAnalysis,ICA,
  Clustering,KMeans,
  HierarchicalClustering,
  SpectralClustering,
  DensityEstimation,
  KernelDensityEstimation,KDE,
  HistogramEstimator,
  NonparametricRegression,
  SmoothingSpline,
  LOESSRelationship,
  HighDimensionalStatistics,
  Regularisation,L1Regularisation,
  L2Regularisation,Lasso,RidgeRegression,
  ElasticNet,
  SparseEstimation,Sparsity,
  VariableSelection,
  MultipleTesting,
  FalseDiscoveryRate,FDR,
  BenjaminiHochbergRelationship,
  EmpiricalBayes,
  ShrinkageEstimator,
  JamesSteinRelationship,
  RandomNumberGeneration,RNG,
  PseudorandomNumberGenerator,PRNG,
  RandomSeed,RandomStream,
  CounterBasedRNG,
  ParallelRandomNumberGeneration,
  DistributionSampling,
  InverseTransformSampling,
  BoxMullerRelationship,
  AliasMethodRelationship,
  Reproducibility,Repeatability,
  Determinism,Nondeterminism,
  NumericalStability,
  FloatingPointArithmetic,
  MachinePrecision,RoundOffError,
  Underflow,Overflow,
  LogSumExp,StableSoftmaxRelationship,
  Conditioning,IllConditioning,
  MatrixComputation,LinearAlgebra,
  MatrixDecomposition,
  CholeskyDecomposition,QRDecomposition,
  SingularValueDecomposition,SVD,
  EigenvalueComputation,
  SparseMatrix,SparseLinearAlgebra,
  ComputationalComplexity,
  TimeComplexity,SpaceComplexity,
  AlgorithmicComplexity,
  StatisticalComplexityRelationship,
  Convergence,EstimatorConvergence,
  MCMCConvergence,OptimisationConvergence,
  NumericalConvergence,
  BurnIn,Warmup,
  Mixing,Autocorrelation,
  IntegratedAutocorrelationTime,
  MonteCarloStandardError,MCSE,
  GelmanRubinStatistic,RHat,
  TracePlotRelationship,
  EffectiveSampleSizeDiagnostic,
  Bias,Variance,MeanSquaredError,MSE,
  StandardError,ConfidenceInterval,
  CredibleInterval,
  HypothesisTesting,PValue,
  TestStatistic,Power,
  TypeIError,TypeIIError,
  CoverageProbability,
  Calibration,PosteriorCalibration,
  SimulationBasedCalibration,SBC,
  PosteriorPredictiveCheck,PPC,
  SensitivityAnalysis,
  RobustStatistics,
  RobustEstimator,HuberLoss,
  MEstimator,
  MissingData,Imputation,
  MultipleImputation,
  ExpectationMaximisationForMissingData,
  DataAugmentation,
  ParallelStatistics,
  DistributedStatistics,
  GPUStatistics,
  HighPerformanceStatisticalComputing,
  BatchComputation,Vectorisation,
  ParallelMonteCarlo,
  DistributedMCMC,
  ParallelBootstrap,
  MapReduceStatisticsRelationship,
  ScientificComputing,
  HighPerformanceComputing,
  Verification,Validation,
  Benchmarking,ReferenceResult,
  Provenance,SoftwareVersion,
  LibraryVersion,Configuration
 ]
 EXCLUDE_AS_PRIMARY = [
  PureTheoreticalStatisticsWithoutComputationalMethod,
  GenericDataProcessingWithoutStatisticalInference,
  DeterministicNumericalAnalysisWithoutStatisticalSemantics,
  MachineLearningWithoutStatisticalComputationContext
 ]
 DISAMBIGUATION_RULE = "When statistical inference, estimation or uncertainty analysis is performed using numerical algorithms, simulation, optimisation or resampling, interpret the context as computational statistics."
}

CORE_CONCEPTS {
 ComputationalStatistics = "Algorithmic and numerical execution of statistical inference."
 MonteCarlo = "Estimation using random samples."
 Bootstrap = "Resampling method used to approximate sampling distributions."
 MCMC = "Simulation method for sampling from difficult probability distributions through Markov chains."
 VariationalInference = "Optimisation-based approximation of probability distributions."
 MaximumLikelihood = "Parameter estimation by maximising likelihood."
 BayesianComputation = "Numerical inference based on posterior distributions."
 NumericalStability = "Reliability of statistical computation under finite precision."
}

RELATIONS {
 ComputationalStatistics --is_child_of--> ComputationalMathematicsOntologies
 ComputationalStatistics --uses--> NumericalMethods
 ComputationalStatistics --uses--> ProbabilityAndStatistics
 MonteCarlo --estimates--> Expectation
 MCMC --samples_from--> PosteriorDistribution
 ImportanceSampling --uses--> ProposalDistribution
 SequentialMonteCarlo --uses--> WeightedParticles
 Bootstrap --approximates--> SamplingDistribution
 PermutationTest --constructs--> NullDistribution
 MaximumLikelihood --uses--> Optimisation
 EM --optimises--> IncompleteDataLikelihood
 VariationalInference --optimises--> ELBO
 ApproximateBayesianComputation --supports--> LikelihoodFreeInference
 CrossValidation --estimates--> PredictivePerformance
 MatrixDecomposition --supports--> StatisticalLinearAlgebra
 Verification --checks--> StatisticalAlgorithmImplementation
 Reproducibility --requires--> SeedConfigurationAndVersions
}

OPERATORS {
 MONTE_CARLO_MEAN(f,X) := (1/N)*sum_i f(X_i)
 LOG_LIKELIHOOD(theta,data) := sum_i log p(data_i|theta)
 MLE(data) := argmax_theta LOG_LIKELIHOOD(theta,data)
 POSTERIOR(theta,data) := likelihood(data|theta)*prior(theta)/evidence(data)
 IMPORTANCE_ESTIMATE(f,w,x) := sum_i w_i*f(x_i)/sum_i w_i
 BOOTSTRAP(data,T,B) := {T(resample_with_replacement(data)) for b in 1..B}
 MH_ACCEPTANCE(x,x_new) := min(1,target(x_new)*proposal(x|x_new)/(target(x)*proposal(x_new|x)))
 ELBO(q) := E_q[log p(data,theta)-log q(theta)]
}

AXIOMS {
 A1 = "Computational statistics combines statistical principles with numerical and algorithmic methods."
 A2 = "Simulation-based estimators introduce Monte Carlo uncertainty."
 A3 = "Optimisation error and statistical estimation error are distinct."
 A4 = "Posterior approximation quality depends on algorithm, convergence and numerical stability."
 A5 = "Resampling approximates sampling behaviour using repeated recomputation."
 A6 = "Random-number generation and seed semantics affect reproducibility."
 A7 = "Finite precision can materially affect likelihoods, matrix computations and tail probabilities."
 A8 = "Computational-statistical results require explicit algorithm, convergence, uncertainty and reproducibility semantics."
}

REASONING {
 R1_CONTEXT = identify(statistical_question,data_structure,model,parameters,uncertainty_goal,computational_budget)
 R2_METHOD = classify(likelihood,Bayesian,resampling,MonteCarlo,MCMC,SMC,variational_or_optimisation_method)
 R3_REPRESENTATION = define(data,parameters,latent_variables,priors,likelihoods_and_constraints)
 R4_NUMERICS = choose(stable_linear_algebra,log_domain_computation,optimisation_and_sampling_algorithms)
 R5_RANDOMNESS = configure(PRNG,seed,independent_streams_and_parallel_randomness)
 R6_EXECUTION = perform(iteration,sampling,resampling,optimisation_or_numerical_integration)
 R7_CONVERGENCE = assess(optimisation_tolerance,MCMC_mixing,ESS,MCSE_or_resampling_stability)
 R8_DIAGNOSTICS = check(bias,variance,coverage,predictive_fit,posterior_checks_and_numerical_failures)
 R9_SCALING = if_large use(vectorisation,parallelism,distributed_or_GPU_computation)
 R10_VALIDATE = compare(reference_cases,simulation_truth,known_results_and_reproducibility)
}

CONSTRAINTS {
 C1 = "Monte Carlo estimates can have high variance when sample sizes are insufficient."
 C2 = "MCMC chains can converge slowly or fail to explore multimodal distributions."
 C3 = "Importance sampling can fail when proposal and target distributions have poor overlap."
 C4 = "Bootstrap methods can be unreliable for some dependent, nonsmooth or boundary problems."
 C5 = "Likelihood evaluation can underflow without log-domain computation."
 C6 = "Ill-conditioned matrices can destabilise statistical estimation."
 C7 = "Variational approximations can underestimate posterior uncertainty."
 C8 = "Optimisation convergence does not guarantee global optimum or inferential validity."
 C9 = "Parallel RNG streams require explicit independence or counter-based design."
 C10 = "Numerically precise output is not sufficient evidence of statistical validity."
}

SEMANTIC_EQUIVALENCE {
 "ComputationalStatistics" <=> "StatisticalComputing"
 "MaximumLikelihoodEstimation" <=> "MLE"
 "MarkovChainMonteCarlo" <=> "MCMC"
 "SequentialMonteCarlo" <=> "SMC"
 "VariationalInference" <=> "VI"
 "ApproximateBayesianComputation" <=> "ABC"
 "EffectiveSampleSize" <=> "ESS"
 "MonteCarloStandardError" <=> "MCSE"
}

VALIDATION {
 V1 = "Statistical model and estimand are explicit."
 V2 = "Algorithm and numerical settings are documented."
 V3 = "Random seed and RNG are documented where stochastic methods are used."
 V4 = "Optimisation or sampling convergence is assessed."
 V5 = "Monte Carlo or resampling uncertainty is quantified where relevant."
 V6 = "Numerical stability is checked for likelihoods and matrix operations."
 V7 = "Simulation, benchmark or known-answer tests are used where feasible."
 V8 = "Predictive or posterior diagnostics are documented where applicable."
 V9 = "Parallel or distributed randomness semantics are documented where used."
 V10 = "Results are reproducible under documented data, seed, software and configuration settings."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 COMPUTATIONAL_STATISTICS_DEFINITION = 0.99
 MONTE_CARLO_REASONING = 0.99
 MCMC_REASONING = 0.99
 BOOTSTRAP_REASONING = 0.99
 LIKELIHOOD_REASONING = 0.99
 VARIATIONAL_REASONING = 0.98
 NUMERICAL_STABILITY_REASONING = 0.99
 POLICY = "Reduce confidence when statistical model, convergence, RNG, sample size, numerical stability, diagnostics or approximation assumptions are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,OntologyGrounding,ComputationalStatisticsReasoning,
  MonteCarloReasoning,BootstrapReasoning,
  MCMCReasoning,ImportanceSamplingReasoning,
  SMCReasoning,LikelihoodReasoning,
  BayesianComputationReasoning,
  VariationalInferenceReasoning,
  EMReasoning,OptimisationReasoning,
  CrossValidationReasoning,
  RandomNumberGenerationReasoning,
  ConvergenceDiagnosticReasoning,
  NumericalStabilityReasoning,
  StatisticalUncertaintyReasoning,
  ReproducibilityReasoning,
  KnowledgeGraphConstruction,RetrievalAugmentation,RecursiveInference
 ]
 NORMALISE {
  "computational statistics" -> ComputationalStatistics
  "statistical computing" -> ComputationalStatistics
  "MLE" -> MaximumLikelihood
  "MCMC" -> MarkovChainMonteCarlo
  "SMC" -> SequentialMonteCarlo
  "VI" -> VariationalInference
  "ABC" -> ApproximateBayesianComputation
  "ESS" -> EffectiveSampleSize
  "MCSE" -> MonteCarloStandardError
 }
}

EXECUTION_RULES {
 E1 = "If statistical inference is carried out through numerical algorithms, simulation, optimisation or resampling, activate ComputationalStatistics."
 E2 = "Define estimand, model, uncertainty target and data structure before selecting computation."
 E3 = "Use numerically stable likelihood and matrix computations."
 E4 = "For stochastic methods, document RNG, seed and stream semantics."
 E5 = "Assess convergence separately from inferential adequacy."
 E6 = "Quantify Monte Carlo or resampling error where material."
 E7 = "Use multiple chains or diagnostics for MCMC where appropriate."
 E8 = "For large workloads, use vector, parallel, distributed or GPU computation with explicit reproducibility semantics."
 E9 = "Validate against known solutions, simulated truth, benchmarks or independent diagnostics."
}

RECURSION {
 ENTRY = ComputationalStatistics
 EXPAND(ComputationalStatistics) -> [
  StatisticalComputing,NumericalStatistics,
  StatisticalInference,StatisticalEstimation,
  StatisticalModel,Parameter,Estimator,
  Likelihood,LogLikelihood,MaximumLikelihood,
  PenalisedLikelihood,BayesianInference,
  PriorDistribution,PosteriorDistribution,
  MonteCarlo,MonteCarloEstimator,
  VarianceReduction,ImportanceSampling,
  RejectionSampling,MarkovChainMonteCarlo,
  MCMC,MetropolisHastings,GibbsSampling,
  HamiltonianMonteCarlo,HMC,NUTS,
  SliceSampling,AdaptiveMCMC,
  SequentialMonteCarlo,SMC,ParticleFilter,
  EffectiveSampleSize,ESS,
  AnnealedImportanceSampling,Bootstrap,
  ParametricBootstrap,BlockBootstrap,
  PermutationTest,Jackknife,
  CrossValidation,KFoldCrossValidation,
  ModelSelection,AIC,BIC,
  ExpectationMaximisation,EM,
  VariationalInference,VI,ELBO,
  LaplaceApproximation,
  ApproximateBayesianComputation,ABC,
  NumericalIntegration,Quadrature,
  QuasiMonteCarlo,QMC,
  NumericalOptimisation,GradientDescent,
  NewtonMethod,BFGS,LBFGS,
  StochasticGradientDescent,
  RootFinding,FisherScoring,IRLS,
  LinearRegression,GLM,MixedModel,
  GaussianProcess,StateSpaceModel,HMM,
  MixtureModel,PCA,Clustering,
  KernelDensityEstimation,HighDimensionalStatistics,
  Regularisation,Lasso,RidgeRegression,
  MultipleTesting,FDR,EmpiricalBayes,
  RandomNumberGeneration,PRNG,
  RandomSeed,ParallelRandomNumberGeneration,
  NumericalStability,FloatingPointArithmetic,
  LogSumExp,Conditioning,
  MatrixComputation,CholeskyDecomposition,
  QRDecomposition,SVD,SparseMatrix,
  ComputationalComplexity,Convergence,
  MCMCConvergence,BurnIn,Mixing,
  Autocorrelation,MCSE,RHat,
  Bias,Variance,MSE,StandardError,
  ConfidenceInterval,CredibleInterval,
  HypothesisTesting,Power,
  SimulationBasedCalibration,
  PosteriorPredictiveCheck,
  SensitivityAnalysis,RobustStatistics,
  MissingData,MultipleImputation,
  ParallelStatistics,DistributedStatistics,
  GPUStatistics,HighPerformanceStatisticalComputing,
  Verification,Validation,Benchmarking,
  Reproducibility,Provenance
 ]
 EXPAND(ComputationalStatisticsFamilies) -> [
  MonteCarloAndSimulationMethods,
  ResamplingMethods,
  BayesianComputationalMethods,
  MCMCandSequentialMonteCarlo,
  LikelihoodAndOptimisationMethods,
  ApproximateInferenceMethods,
  HighDimensionalStatisticalComputation,
  ParallelAndDistributedStatisticalComputing
 ]
 STOP_CONDITION = "Stop when model, estimand, algorithm, randomness, numerical method, convergence, diagnostics, uncertainty and reproducibility are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Computational Mathematics Ontologies"
 RELATED_ONTOLOGIES = [
  "Probability and Statistics Ontologies",
  "Monte Carlo Simulation Ontology",
  "Random Process Ontology",
  "Optimisation Computation",
  "Numerical Methods Ontology 2",
  "Numerical Linear Algebra Ontology 2",
  "Matrix Computation Ontology",
  "Sparse Matrix Ontology 2",
  "Numerical Error Ontology",
  "Numerical Stability Ontology 2",
  "Numerical Convergence Ontology",
  "Floating Point Arithmetic Ontology",
  "High Performance Computing Ontology 2",
  "Parallel Computing Ontology",
  "Distributed Computing Ontology",
  "GPU Computing Ontology"
 ]
 ROUTE_RULES = [
  "Computational Statistics -> Probability and Statistics for parent statistical semantics",
  "Computational Statistics -> Monte Carlo Simulation for simulation-based estimation",
  "Computational Statistics -> Random Process for MCMC and stochastic-state semantics",
  "Computational Statistics -> Optimisation Computation for likelihood and variational optimisation",
  "Computational Statistics -> Numerical Methods for quadrature, root finding and numerical approximation",
  "Computational Statistics -> Numerical Linear Algebra for regression, covariance and latent-variable computation",
  "Computational Statistics -> Matrix Computation for dense statistical kernels",
  "Computational Statistics -> Sparse Matrix for high-dimensional sparse models",
  "Computational Statistics -> Numerical Error for approximation and finite-precision error",
  "Computational Statistics -> Numerical Stability for robust likelihood and matrix computation",
  "Computational Statistics -> Numerical Convergence for iterative and sampling convergence",
  "Computational Statistics -> Floating Point Arithmetic for underflow, overflow and accumulation effects",
  "Computational Statistics -> High Performance Computing for large-scale statistical workloads",
  "Computational Statistics -> Parallel, Distributed and GPU Computing for scalable execution"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible statistical algorithms, diagnostics or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve statistical-model-versus-computational-algorithm distinction.",
  "Preserve sampling-error-versus-numerical-error distinction.",
  "Preserve optimisation-convergence-versus-statistical-validity distinction.",
  "Preserve prior-versus-likelihood-versus-posterior distinction.",
  "Preserve resampling-versus-parametric simulation distinction.",
  "Preserve exact-versus-approximate inference distinction.",
  "Preserve deterministic-versus-stochastic algorithm distinction.",
  "Preserve machine-readable computational-statistics semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Computational statistics applies numerical algorithms, simulation and optimisation to statistical estimation, inference and uncertainty analysis."
 MONTE_CARLO_SUMMARY = "Monte Carlo methods approximate expectations and distributions through random sampling."
 RESAMPLING_SUMMARY = "Bootstrap, jackknife and permutation methods approximate sampling behaviour through repeated recomputation."
 BAYESIAN_SUMMARY = "MCMC, SMC, importance sampling and variational methods approximate Bayesian posterior quantities."
 LIKELIHOOD_SUMMARY = "Maximum-likelihood and EM-style methods use numerical optimisation and linear algebra to fit statistical models."
 DIAGNOSTIC_SUMMARY = "Convergence, effective sample size, Monte Carlo error, predictive checks and numerical stability determine computational reliability."
 VALIDATION_SUMMARY = "Computational-statistical results should be checked through known-answer tests, simulation, diagnostics, uncertainty estimates, numerical stability and reproducibility."
}

FLOW {
 INPUT
 -> detect_computational_statistics_context
 -> identify_statistical_question_data_structure_model_estimand_uncertainty_goal_and_computational_budget
 -> classify_likelihood_Bayesian_resampling_Monte_Carlo_MCMC_SMC_variational_or_optimisation_method
 -> define_parameters_latent_variables_priors_likelihood_constraints_and_target_quantities
 -> choose_numerically_stable_linear_algebra_log_domain_optimisation_sampling_or_integration_algorithms
 -> configure_PRNG_seed_streams_and_parallel_randomness_if_stochastic
 -> execute_iterations_samples_resamples_optimisation_or_numerical_integration
 -> assess_optimisation_tolerance_sampling_mixing_ESS_MCSE_resampling_stability_or_numerical_convergence
 -> check_bias_variance_coverage_predictive_fit_posterior_diagnostics_and_numerical_failures
 -> if_large_vectorise_parallelise_distribute_or_accelerate_computation
 -> verify_against_simulated_truth_known_results_reference_implementations_or_benchmarks
 -> quantify_statistical_computational_and_numerical_uncertainty
 -> document_data_seed_software_versions_configuration_and_provenance
 -> route_to_related_probability_statistics_Monte_Carlo_random_process_optimisation_numerical_methods_linear_algebra_matrix_sparse_error_stability_convergence_floating_point_HPC_parallel_distributed_and_GPU_ontologies
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Computational Statistics ontology defines algorithmic and numerical methods for statistical inference and estimation. It supports likelihood and maximum-likelihood computation, Bayesian inference, Monte Carlo estimators and variance reduction, importance and rejection sampling, MCMC including Metropolis-Hastings, Gibbs, HMC, NUTS and Langevin methods, sequential Monte Carlo and particle filtering, bootstrap, block bootstrap, permutation and jackknife methods, cross-validation and model selection, EM, variational inference, Laplace and likelihood-free inference, numerical quadrature and quasi-Monte Carlo, optimisation, Fisher scoring and IRLS, regression, mixed, latent-variable, Gaussian-process, state-space and mixture models, nonparametric and high-dimensional statistics, regularisation, multiple testing, RNGs, numerical stability, matrix decompositions, convergence and MCMC diagnostics, Monte Carlo standard error, simulation-based calibration, posterior predictive checking, missing data, parallel, distributed and GPU statistical computation, verification, validation and reproducibility."
}

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

}

END_FLOW