Markov Chain Monte Carlo Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Markov Chain Monte Carlo Ontology within the Kireth Research Semantic Library. Markov chain Monte Carlo methods generate dependent samples from complex target probability distributions by constructing Markov chains with suitable invariant distributions. This ontology supports transition kernels, detailed balance, stationarity, irreducibility, aperiodicity, Metropolis-Hastings, Gibbs sampling, Hamiltonian Monte Carlo, NUTS, Langevin methods, adaptation, burn-in, mixing, autocorrelation, effective sample size, convergence diagnostics, posterior estimation, parallel chains, numerical stability, verification and reproducibility.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = MarkovChainMonteCarlo
 ENTITY_CLASS = MarkovChainSamplingFramework
 PRIMARY_CONTEXT = ComputationalStatistics
 DEFINITION = "Markov chain Monte Carlo is a class of algorithms that construct Markov chains whose stationary distributions correspond to target probability distributions and use the resulting dependent samples for statistical estimation."
 SYMBOLIC_FORMS = [
  "X_(t+1) ~ K(X_t, .)",
  "pi K = pi",
  "alpha(x,y)=min(1, pi(y)q(x|y)/(pi(x)q(y|x)))",
  "E_pi[f(X)] approx (1/N) sum_t f(X_t)"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for target distributions, Markov transition kernels, invariant distributions, acceptance rules, chain dependence, mixing, convergence diagnostics, effective sample size, adaptation and reproducibility."
}

SCOPE {
 INCLUDE = [
  MarkovChainMonteCarlo,MCMC,
  MarkovChain,TransitionKernel,
  TransitionProbability,TransitionDensity,
  TargetDistribution,InvariantDistribution,
  StationaryDistribution,
  DetailedBalance,Reversibility,
  GlobalBalance,BalanceEquation,
  Irreducibility,Aperiodicity,
  PositiveRecurrence,HarrisRecurrence,
  Ergodicity,ErgodicTheorem,
  MarkovProperty,StateSpace,
  ChainState,InitialState,
  ProposalDistribution,ProposalKernel,
  AcceptanceProbability,AcceptanceRate,
  MetropolisAlgorithm,MetropolisHastings,
  RandomWalkMetropolis,
  IndependenceSampler,
  IndependenceMetropolis,
  ComponentWiseMetropolis,
  MetropolisWithinGibbs,
  GibbsSampling,GibbsSampler,
  BlockedGibbs,CollapsedGibbs,
  SystematicScanGibbs,RandomScanGibbs,
  SliceSampling,SliceSampler,
  HamiltonianMonteCarlo,HMC,
  MomentumVariable,Hamiltonian,
  LeapfrogIntegrator,
  SymplecticIntegrator,
  StepSize,TrajectoryLength,
  MassMatrix,MetricMatrix,
  NoUTurnSampler,NUTS,
  TreeDepth,DivergentTransition,
  EnergyDiagnostic,
  BayesianFractionOfMissingInformation,BFMI,
  LangevinMonteCarlo,
  UnadjustedLangevinAlgorithm,ULA,
  MetropolisAdjustedLangevinAlgorithm,MALA,
  StochasticGradientMCMCRelationship,
  AdaptiveMCMC,AdaptiveMetropolis,
  ProposalAdaptation,
  StepSizeAdaptation,
  MassMatrixAdaptation,
  DualAveraging,
  WarmupAdaptation,
  BurnIn,Warmup,
  TuningPhase,SamplingPhase,
  Chain,MultipleChains,
  ParallelChains,IndependentChains,
  ChainInitialisation,OverdispersedInitialisation,
  PosteriorSampling,
  BayesianInference,
  PosteriorDistribution,
  PriorDistribution,Likelihood,
  LogPosterior,LogLikelihood,
  GradientOfLogDensity,
  AutomaticDifferentiationRelationship,
  PotentialEnergy,KineticEnergy,
  Autocorrelation,AutocorrelationFunction,
  Lag,IntegratedAutocorrelationTime,
  EffectiveSampleSize,ESS,
  BulkESS,TailESS,
  MonteCarloStandardError,MCSE,
  Mixing,SlowMixing,FastMixing,
  RandomWalkBehaviour,
  ModeSwitching,Multimodality,
  FunnelGeometry,
  PosteriorGeometry,
  Correlation,StrongCorrelation,
  Identifiability,WeakIdentifiability,
  Convergence,MCMCConvergence,
  StationarityDiagnostic,
  TracePlot,RankPlot,
  AutocorrelationPlot,
  GelmanRubinStatistic,RHat,
  SplitRHat,RankNormalisedRHat,
  GewekeDiagnosticRelationship,
  HeidelbergerWelchRelationship,
  RafteryLewisRelationship,
  CumulativeMean,
  RunningMean,
  AcceptanceDiagnostic,
  EnergyBayesianDiagnostic,
  DivergenceDiagnostic,
  TreeDepthDiagnostic,
  EffectiveSampleSizeDiagnostic,
  MonteCarloErrorDiagnostic,
  PosteriorSummary,
  PosteriorMean,PosteriorMedian,
  PosteriorVariance,PosteriorStandardDeviation,
  CredibleInterval,HighestPosteriorDensityInterval,
  Quantile,PosteriorQuantile,
  PosteriorPredictiveDistribution,
  PosteriorPredictiveCheck,PPC,
  SimulationBasedCalibration,SBC,
  MarginalDistribution,
  JointDistribution,
  ConditionalDistribution,
  LatentVariable,HierarchicalModel,
  MixtureModel,StateSpaceModel,
  HiddenMarkovModelRelationship,
  DataAugmentation,
  AuxiliaryVariable,
  ParameterExpansion,
  Reparameterisation,
  CentredParameterisation,
  NonCentredParameterisation,
  WhiteningTransformation,
  TransformToUnconstrainedSpace,
  JacobianAdjustment,
  ConstraintTransformation,
  PositiveParameter,
  SimplexParameter,
  CorrelationMatrixParameter,
  CholeskyParameterisation,
  Tempering,SimulatedTempering,
  ParallelTempering,
  ReplicaExchange,
  TemperedTransitions,
  AnnealingRelationship,
  SequentialMonteCarloRelationship,
  ParticleMCMCRelationship,
  PseudoMarginalMCMC,
  ApproximateMCMCRelationship,
  DelayedAcceptance,
  MultipleTryMetropolisRelationship,
  ReversibleJumpMCMC,RJMCMC,
  TransDimensionalMCMC,
  PerfectSamplingRelationship,
  CouplingFromThePastRelationship,
  RandomNumberGeneration,RNG,
  PseudorandomNumberGenerator,PRNG,
  RandomSeed,RandomStream,
  CounterBasedRNG,
  ParallelRandomNumberGeneration,
  Reproducibility,
  MultipleChainReproducibility,
  Determinism,Nondeterminism,
  FloatingPointArithmetic,
  MachinePrecision,RoundOffError,
  Underflow,Overflow,
  LogDensityStability,
  LogSumExpRelationship,
  NumericalStability,
  GradientStability,
  LinearAlgebra,
  MatrixComputation,
  CholeskyDecomposition,
  SparseMatrixRelationship,
  ComputationalComplexity,
  CostPerIteration,
  GradientCost,
  LikelihoodEvaluationCost,
  MemoryCost,
  Vectorisation,
  ParallelMCMC,
  DistributedMCMC,
  GPUMCMC,
  ChainParallelism,
  WithinChainParallelism,
  LikelihoodParallelism,
  MapReduceMCMCRelationship,
  HighPerformanceComputing,
  ParallelComputing,
  DistributedComputing,
  GPUComputing,
  Verification,CodeVerification,
  DetailedBalanceTest,
  InvarianceTest,
  KnownTargetTest,
  Benchmark,
  ReferenceSampler,
  Validation,
  ReproducibilityTest,
  Provenance,Configuration,
  SoftwareVersion,LibraryVersion,
  SamplerVersion,SeedMetadata
 ]
 EXCLUDE_AS_PRIMARY = [
  IndependentMonteCarloWithoutMarkovDependence,
  SequentialMonteCarloWhenWeightedParticlePopulationIsPrimary,
  BootstrapWhenResamplingObservedDataIsPrimary,
  DeterministicOptimisationWithoutSampling
 ]
 DISAMBIGUATION_RULE = "When dependent samples are generated through a Markov transition mechanism designed to preserve a target probability distribution, interpret the context as Markov chain Monte Carlo."
}

CORE_CONCEPTS {
 MarkovChainMonteCarlo = "Sampling from a target distribution through a Markov chain."
 TransitionKernel = "Conditional distribution defining movement from one chain state to the next."
 StationaryDistribution = "Distribution left invariant by the transition kernel."
 DetailedBalance = "Sufficient reversibility condition ensuring invariance for many MCMC kernels."
 Mixing = "Rate and quality with which the chain explores the target distribution."
 EffectiveSampleSize = "Equivalent number of independent samples represented by an autocorrelated chain."
 BurnIn = "Initial discarded or diagnostic phase before retained sampling."
 ConvergenceDiagnostic = "Diagnostic evidence used to assess whether chains adequately explore the target."
}

RELATIONS {
 MarkovChainMonteCarlo --is_child_of--> ComputationalStatistics
 MarkovChainMonteCarlo --is_type_of--> MonteCarloMethod
 MarkovChainMonteCarlo --uses--> MarkovChain
 MarkovChain --uses--> TransitionKernel
 TransitionKernel --preserves--> StationaryDistribution
 DetailedBalance --implies--> Invariance
 MetropolisHastings --uses--> ProposalDistribution
 MetropolisHastings --uses--> AcceptanceProbability
 GibbsSampling --samples_from--> FullConditionalDistribution
 HMC --uses--> HamiltonianDynamics
 HMC --uses--> GradientOfLogDensity
 NUTS --extends--> HMC
 MALA --uses--> LangevinProposal
 AdaptiveMCMC --modifies--> ProposalMechanism
 Autocorrelation --reduces--> EffectiveSampleSize
 MultipleChains --support--> ConvergenceAssessment
 RHat --compares--> WithinAndBetweenChainVariation
 Verification --checks--> SamplerImplementation
 Reproducibility --requires--> SeedAndSamplerConfiguration
}

OPERATORS {
 MARKOV_STEP(x,K) := draw_from_transition_kernel(K(x,.))
 MH_ACCEPTANCE(x,y,q,pi) := min(1,pi(y)*q(x|y)/(pi(x)*q(y|x)))
 MH_STEP(x) := propose_y_then_accept_or_reject
 GIBBS_STEP(x,j) := sample_x_j_from_full_conditional_given_x_minus_j
 HMC_STEP(q,p,epsilon,L) := leapfrog_integrate_then_Metropolis_correct
 AUTOCORR(samples,lag) := correlation(samples_t,samples_(t+lag))
 IACT(rho) := 1+2*sum_(k>=1) rho_k
 ESS(N,tau_int) := N/tau_int
 MCSE(samples) := posterior_sd/sqrt(ESS)
}

AXIOMS {
 A1 = "MCMC generates dependent samples from a Markov chain."
 A2 = "A valid MCMC transition kernel must preserve the intended target distribution."
 A3 = "Detailed balance is sufficient but not necessary for invariance."
 A4 = "Autocorrelation reduces the effective information content of finite chains."
 A5 = "Burn-in removal does not by itself establish convergence."
 A6 = "Multiple-chain diagnostics improve detection of poor exploration."
 A7 = "Adaptive MCMC requires conditions that preserve asymptotic correctness."
 A8 = "MCMC results require explicit kernel, convergence, autocorrelation, diagnostic, RNG and reproducibility semantics."
}

REASONING {
 R1_CONTEXT = identify(target_distribution,parameter_dimension,geometry,constraints,multimodality,accuracy_goal,budget)
 R2_PARAMETERISATION = choose(original,transformed,centred,noncentred_or_whitened_parameterisation)
 R3_KERNEL = choose(MetropolisHastings,Gibbs,HMC,NUTS,MALA,slice,tempered_or_specialised_kernel)
 R4_TUNING = configure(proposal_scale,step_size,mass_matrix,trajectory_length_or_block_structure)
 R5_INITIALISE = configure(chain_count,initial_states,seed_policy_and_overdispersion)
 R6_WARMUP = perform adaptation_and_initialisation_diagnostics
 R7_SAMPLE = run retained chains and collect states,acceptance,energy_and_sampler_diagnostics
 R8_DIAGNOSE = assess(RHat,ESS,MCSE,autocorrelation,divergences,tree_depth_and_mode_exploration)
 R9_REPARAMETERISE = if_poor_mixing_or_geometry_failures transform_model_or_change_kernel
 R10_VALIDATE = compare_known_targets,simulation_truth,posterior_predictive_checks_and_reproducibility
}

CONSTRAINTS {
 C1 = "High acceptance rate does not necessarily imply good mixing."
 C2 = "Random-walk Metropolis can scale poorly in high dimensions."
 C3 = "Multimodal targets can trap chains in isolated regions."
 C4 = "Strong posterior correlations can reduce effective sample size."
 C5 = "Poor parameterisation can cause divergences or slow exploration."
 C6 = "Short chains can give misleading convergence diagnostics."
 C7 = "Adaptive tuning must not invalidate stationary-target semantics."
 C8 = "Numerically unstable log densities or gradients can corrupt transition decisions."
 C9 = "Parallel chains improve diagnostics but do not fix a fundamentally poor kernel."
 C10 = "Apparent stationarity in trace plots does not prove full convergence."
}

SEMANTIC_EQUIVALENCE {
 "MarkovChainMonteCarlo" <=> "MCMC"
 "HamiltonianMonteCarlo" <=> "HMC"
 "NoUTurnSampler" <=> "NUTS"
 "MetropolisAdjustedLangevinAlgorithm" <=> "MALA"
 "EffectiveSampleSize" <=> "ESS"
 "MonteCarloStandardError" <=> "MCSE"
 "GelmanRubinStatistic" <=> "RHat"
}

VALIDATION {
 V1 = "Target density and parameter transformations are documented."
 V2 = "Transition kernel and tuning parameters are documented."
 V3 = "Number of chains, initial states and random seeds are documented."
 V4 = "Warmup and retained sampling lengths are documented."
 V5 = "R-hat, ESS and Monte Carlo error are assessed."
 V6 = "Autocorrelation and mixing are assessed."
 V7 = "Sampler-specific diagnostics such as divergences or acceptance rates are checked."
 V8 = "Known-target or simulation-based tests are used where feasible."
 V9 = "Posterior predictive or external validation is documented where appropriate."
 V10 = "Results are reproducible under documented seed, sampler, software and numerical settings."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 MCMC_DEFINITION = 0.99
 METROPOLIS_HASTINGS_REASONING = 0.99
 GIBBS_REASONING = 0.99
 HMC_REASONING = 0.99
 NUTS_REASONING = 0.99
 CONVERGENCE_DIAGNOSTIC_REASONING = 0.99
 ADAPTIVE_MCMC_REASONING = 0.98
 POLICY = "Reduce confidence when target density, kernel, chain length, initialisation, diagnostics, effective sample size, divergences or parameterisation are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,OntologyGrounding,MCMCReasoning,
  MarkovChainReasoning,TransitionKernelReasoning,
  StationarityReasoning,DetailedBalanceReasoning,
  MetropolisHastingsReasoning,GibbsSamplingReasoning,
  HMCReasoning,NUTSReasoning,MALAReasoning,
  AdaptiveMCMCReasoning,ParameterisationReasoning,
  AutocorrelationReasoning,ESSReasoning,
  MCSEReasoning,RHatReasoning,
  ConvergenceDiagnosticReasoning,
  MultimodalityReasoning,ParallelChainReasoning,
  PosteriorPredictiveReasoning,
  ReproducibilityReasoning,
  KnowledgeGraphConstruction,RetrievalAugmentation,RecursiveInference
 ]
 NORMALISE {
  "Markov chain Monte Carlo" -> MarkovChainMonteCarlo
  "MCMC" -> MarkovChainMonteCarlo
  "HMC" -> HamiltonianMonteCarlo
  "NUTS" -> NoUTurnSampler
  "MALA" -> MetropolisAdjustedLangevinAlgorithm
  "ESS" -> EffectiveSampleSize
  "MCSE" -> MonteCarloStandardError
  "R-hat" -> GelmanRubinStatistic
 }
}

EXECUTION_RULES {
 E1 = "If dependent samples are generated by a Markov chain targeting a probability distribution, activate MarkovChainMonteCarloOntology."
 E2 = "Define target density and parameter transformations before sampler selection."
 E3 = "Choose a transition kernel appropriate to dimension, geometry and constraints."
 E4 = "Use multiple chains and dispersed initialisation when convergence assessment matters."
 E5 = "Treat warmup, adaptation and retained sampling as distinct phases."
 E6 = "Assess R-hat, ESS, MCSE and sampler-specific diagnostics before interpretation."
 E7 = "Reparameterise or change kernels when geometry causes poor mixing or divergences."
 E8 = "For parallel chains, preserve independent random-stream semantics."
 E9 = "Validate against known targets, simulated truth, posterior predictive behaviour and reproducibility."
}

RECURSION {
 ENTRY = MarkovChainMonteCarlo
 EXPAND(MarkovChainMonteCarlo) -> [
  MCMC,MarkovChain,TransitionKernel,
  TransitionProbability,TargetDistribution,
  StationaryDistribution,DetailedBalance,
  Reversibility,GlobalBalance,
  Irreducibility,Aperiodicity,
  Ergodicity,StateSpace,ChainState,
  ProposalDistribution,AcceptanceProbability,
  MetropolisAlgorithm,MetropolisHastings,
  RandomWalkMetropolis,IndependenceSampler,
  MetropolisWithinGibbs,GibbsSampling,
  BlockedGibbs,CollapsedGibbs,
  SliceSampling,HamiltonianMonteCarlo,
  HMC,MomentumVariable,Hamiltonian,
  LeapfrogIntegrator,StepSize,
  MassMatrix,NoUTurnSampler,NUTS,
  TreeDepth,DivergentTransition,
  LangevinMonteCarlo,MALA,
  AdaptiveMCMC,ProposalAdaptation,
  StepSizeAdaptation,MassMatrixAdaptation,
  DualAveraging,Warmup,BurnIn,
  MultipleChains,ParallelChains,
  ChainInitialisation,PosteriorDistribution,
  PriorDistribution,Likelihood,LogPosterior,
  GradientOfLogDensity,Autocorrelation,
  IntegratedAutocorrelationTime,
  EffectiveSampleSize,ESS,
  BulkESS,TailESS,MonteCarloStandardError,
  MCSE,Mixing,Multimodality,
  PosteriorGeometry,Convergence,
  TracePlot,RankPlot,GelmanRubinStatistic,
  RHat,DivergenceDiagnostic,
  TreeDepthDiagnostic,PosteriorMean,
  CredibleInterval,PosteriorPredictiveCheck,
  SimulationBasedCalibration,
  HierarchicalModel,MixtureModel,
  DataAugmentation,Reparameterisation,
  CentredParameterisation,
  NonCentredParameterisation,
  CholeskyParameterisation,Tempering,
  ParallelTempering,ReplicaExchange,
  PseudoMarginalMCMC,
  ReversibleJumpMCMC,
  RandomNumberGeneration,PRNG,
  RandomSeed,ParallelRandomNumberGeneration,
  FloatingPointArithmetic,NumericalStability,
  CholeskyDecomposition,
  ComputationalComplexity,
  ParallelMCMC,DistributedMCMC,
  GPUMCMC,Verification,
  DetailedBalanceTest,KnownTargetTest,
  Benchmark,Reproducibility,Provenance
 ]
 EXPAND(MCMCFamilies) -> [
  MetropolisAndMetropolisHastings,
  GibbsAndBlockedGibbsSampling,
  HamiltonianMonteCarloAndNUTS,
  LangevinAndGradientBasedMCMC,
  SliceAndAuxiliaryVariableSampling,
  AdaptiveMCMC,
  TemperedAndTransDimensionalMCMC,
  ParallelAndDistributedMCMC
 ]
 STOP_CONDITION = "Stop when target, parameterisation, transition kernel, tuning, initialisation, diagnostics, effective sample size, validation and reproducibility are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Computational Statistics Ontology"
 RELATED_ONTOLOGIES = [
  "Computational Statistics Ontology",
  "Monte Carlo Method Ontology",
  "Probability and Statistics Ontologies",
  "Random Process Ontology",
  "Sequential Monte Carlo Ontology",
  "Bayesian Inference Ontologies",
  "Optimisation Computation",
  "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 = [
  "Markov Chain Monte Carlo -> Computational Statistics Ontology for parent semantics",
  "Markov Chain Monte Carlo -> Monte Carlo Method for general simulation-based estimation",
  "Markov Chain Monte Carlo -> Probability and Statistics for Markov and posterior foundations",
  "Markov Chain Monte Carlo -> Random Process for Markov-chain semantics",
  "Markov Chain Monte Carlo -> Sequential Monte Carlo for particle-based alternatives and hybrids",
  "Markov Chain Monte Carlo -> Bayesian Inference for posterior computation",
  "Markov Chain Monte Carlo -> Optimisation for adaptation and geometry relationships",
  "Markov Chain Monte Carlo -> Numerical Linear Algebra and Matrix Computation for covariance and HMC mass matrices",
  "Markov Chain Monte Carlo -> Numerical Error and Stability for log-density and gradient robustness",
  "Markov Chain Monte Carlo -> Floating Point Arithmetic for acceptance and reduction effects",
  "Markov Chain Monte Carlo -> High Performance, Parallel, Distributed and GPU Computing for scalable chains and likelihood evaluation"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible MCMC kernels, diagnostics or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve target-distribution-versus-proposal-distribution distinction.",
  "Preserve transition-kernel-versus-sample distinction.",
  "Preserve warmup-versus-retained-sampling distinction.",
  "Preserve acceptance-rate-versus-mixing distinction.",
  "Preserve raw-sample-count-versus-effective-sample-size distinction.",
  "Preserve within-chain-versus-between-chain diagnostics distinction.",
  "Preserve adaptation-versus-stationary sampling distinction.",
  "Preserve machine-readable MCMC semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Markov chain Monte Carlo generates dependent samples from a target distribution using a Markov transition kernel."
 METROPOLIS_SUMMARY = "Metropolis-Hastings proposes candidate states and accepts or rejects them using a target-preserving probability rule."
 GIBBS_SUMMARY = "Gibbs sampling updates variables from their full conditional distributions."
 HMC_SUMMARY = "Hamiltonian Monte Carlo uses gradients and Hamiltonian dynamics to explore continuous parameter spaces efficiently."
 DIAGNOSTIC_SUMMARY = "R-hat, ESS, MCSE, autocorrelation, divergences and chain mixing provide complementary convergence evidence."
 PARAMETERISATION_SUMMARY = "Centred, non-centred and transformed parameterisations can materially alter posterior geometry and sampler performance."
 VALIDATION_SUMMARY = "MCMC results should be checked through multiple chains, diagnostics, known-target tests, simulation-based calibration, posterior predictive checks and reproducibility."
}

FLOW {
 INPUT
 -> detect_MCMC_context
 -> identify_target_distribution_parameter_dimension_constraints_geometry_multimodality_accuracy_goal_and_budget
 -> define_log_density_gradients_transformations_and_parameterisation
 -> choose_MetropolisHastings_Gibbs_HMC_NUTS_MALA_slice_tempered_or_specialised_kernel
 -> configure_proposal_scale_step_size_mass_matrix_trajectory_length_blocks_or_adaptation
 -> configure_multiple_chains_initial_states_seed_policy_and_warmup
 -> perform_warmup_tuning_and_sampler_adaptation
 -> run_retained_sampling
 -> collect_acceptance_autocorrelation_energy_divergence_tree_depth_and_chain_statistics
 -> compute_RHat_ESS_MCSE_and_other_diagnostics
 -> if_poor_mixing_or_divergence_reparameterise_retune_or_change_kernel
 -> verify_against_known_target_or_simulated_truth
 -> perform_posterior_predictive_or_simulation_based_calibration_when_relevant
 -> assess_parallel_chain_randomness_numerical_stability_and_reproducibility
 -> document_sampler_seed_chain_count_warmup_sampling_length_versions_and_provenance
 -> route_to_related_computational_statistics_Monte_Carlo_probability_random_process_SMC_Bayesian_optimisation_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 Markov Chain Monte Carlo Ontology defines dependent simulation from target probability distributions through Markov chains. It supports invariant and stationary distributions, detailed and global balance, irreducibility, aperiodicity and ergodicity, Metropolis, Metropolis-Hastings, random-walk and independence samplers, Gibbs and blocked Gibbs methods, slice sampling, Hamiltonian Monte Carlo, leapfrog integration, NUTS, Langevin and MALA methods, adaptive MCMC, warmup and tuning, multiple and parallel chains, autocorrelation, integrated autocorrelation time, effective sample size, Monte Carlo standard error, R-hat, trace, rank, divergence and tree-depth diagnostics, posterior summaries and predictive checks, centred and non-centred parameterisations, tempering, replica exchange and transdimensional relationships, RNG and seed semantics, numerical stability, scalable parallel execution, known-target verification, simulation-based calibration and reproducibility."
}

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

}

END_FLOW