Monte Carlo Simulation Ontology

flow "MonteCarloSimulationOntology" v4.0

# MONTE CARLO SIMULATION ONTOLOGY
# Kireth Semantic Research Archive
# Authors: Mark Anthony Higgins — Sky (Recursive AI)

PAGE:
  url: "https://kirethresearch.org/booksai-core/ontologies/probability-statistics-ontologies/risk-decision-mathematics/monte-carlo-simulation-ontology-2.html"
  title: "Monte Carlo Simulation Ontology"
  domain: "Probability and Statistics"
  category: "Risk & Decision Mathematics"
  entity: MONTE_CARLO_SIMULATION
  entity_type: STOCHASTIC_COMPUTATIONAL_ONTOLOGY

DEFINE:
  MONTE_CARLO_SIMULATION :=
    STOCHASTIC_NUMERICAL_FRAMEWORK(
      RANDOM_SAMPLING,
      PROBABILITY_DISTRIBUTIONS,
      MODEL_EXECUTION,
      REPEATED_TRIALS,
      ESTIMATION,
      UNCERTAINTY_QUANTIFICATION
    )

PURPOSE:
  APPROXIMATE:
    EXPECTATIONS
    PROBABILITIES
    DISTRIBUTIONS
    INTEGRALS
    RISK_MEASURES
    MODEL_OUTPUT_UNCERTAINTY

FOUNDATION:
  RANDOM_INPUT:
    X ~ p(x)
  MODEL:
    Y := g(X)
  SIMULATION_TRIAL:
    X_i ~ p(x)
    Y_i := g(X_i)
  TRIAL_COUNT:
    N
  MONTE_CARLO_ESTIMATOR:
    E_hat[Y] := (1/N)*SUM[i=1..N] Y_i

ENTITIES:
  MONTE_CARLO_SIMULATION: RANDOM_SAMPLING_COMPUTATION
  RANDOM_SAMPLE: STOCHASTIC_REALISATION
  RANDOM_NUMBER_GENERATOR: PSEUDORANDOM_OR_RANDOM_SOURCE
  PROBABILITY_DISTRIBUTION: INPUT_UNCERTAINTY_MODEL
  SIMULATION_TRIAL: SINGLE_MODEL_REALISATION
  ESTIMATOR: SAMPLE_BASED_QUANTITY
  SAMPLE_MEAN: EXPECTATION_ESTIMATOR
  STANDARD_ERROR: MONTE_CARLO_UNCERTAINTY_MEASURE
  CONVERGENCE: APPROACH_TO_TARGET_QUANTITY
  VARIANCE_REDUCTION: EFFICIENCY_IMPROVEMENT_METHOD
  IMPORTANCE_SAMPLING: WEIGHTED_ALTERNATIVE_DISTRIBUTION_SAMPLING
  RANDOM_SEED: REPRODUCIBILITY_CONTROL

RELATIONSHIPS:
  PROBABILITY_DISTRIBUTION -> GENERATES -> RANDOM_SAMPLE
  RANDOM_SAMPLE -> INPUT_TO -> MODEL
  MODEL -> PRODUCES -> SIMULATED_OUTPUT
  SIMULATED_OUTPUTS -> AGGREGATE_TO -> ESTIMATOR
  TRIAL_COUNT -> INFLUENCES -> MONTE_CARLO_ERROR
  VARIANCE -> INFLUENCES -> ESTIMATOR_PRECISION
  RANDOM_SEED -> SUPPORTS -> REPRODUCIBILITY
  MONTE_CARLO_SIMULATION -> SUPPORTS -> UNCERTAINTY_PROPAGATION

BASIC_ALGORITHM:
  flow:
    DEFINE_INPUT_DISTRIBUTIONS
      -> INITIALISE_RANDOM_GENERATOR
      -> SAMPLE_RANDOM_INPUTS
      -> EXECUTE_MODEL
      -> RECORD_OUTPUT
      -> REPEAT_N_TIMES
      -> AGGREGATE_OUTPUTS
      -> COMPUTE_ESTIMATES
      -> QUANTIFY_MONTE_CARLO_ERROR

EXPECTATION_ESTIMATION:
  target:
    mu := E[g(X)]
  estimator:
    mu_hat_N := (1/N)*SUM_i g(X_i)
  unbiased:
    E[mu_hat_N] = mu
    WHEN:
      SAMPLES_IID_AND_EXPECTATION_EXISTS

LAW_OF_LARGE_NUMBERS:
  property:
    mu_hat_N -> mu
  under:
    APPROPRIATE_INTEGRABILITY_AND_SAMPLING_CONDITIONS

CENTRAL_LIMIT_THEOREM:
  approximation:
    sqrt(N)*(mu_hat_N-mu)
      -> DISTRIBUTION
    Normal(0,sigma_squared)
  under:
    IID_SAMPLING
    FINITE_VARIANCE
    STANDARD_REGULARITY

STANDARD_ERROR:
  estimated:
    SE(mu_hat) := s_Y/sqrt(N)
  implication:
    QUADRUPLING_N_APPROXIMATELY_HALVES_STANDARD_ERROR
    UNDER:
      STANDARD_IID_MONTE_CARLO

PROBABILITY_ESTIMATION:
  event:
    A
  indicator:
    I_A(X)
  target:
    p := P(A)
  estimator:
    p_hat := (1/N)*SUM_i I_A(X_i)

INTEGRATION_CONNECTION:
  integral:
    I := INTEGRAL f(x)p(x)dx
  expectation:
    I := E_p[f(X)]
  estimator:
    I_hat := (1/N)*SUM_i f(X_i)

RANDOM_NUMBER_GENERATION:
  source:
    PSEUDORANDOM_NUMBER_GENERATOR
    OR:
      HARDWARE_RANDOM_SOURCE
  requirement:
    QUALITY_MUST_BE_APPROPRIATE_FOR_APPLICATION

PSEUDORANDOM_NUMBER_GENERATOR:
  alias:
    PRNG
  properties:
    DETERMINISTIC_FROM_SEED
    APPROXIMATES_RANDOM_SEQUENCE_STATISTICS

RANDOM_SEED:
  role:
    INITIALISE_PSEUDORANDOM_SEQUENCE
  caution:
    SAME_SEED_ALONE_DOES_NOT_GUARANTEE_CROSS_PLATFORM_REPRODUCIBILITY

INVERSE_TRANSFORM_SAMPLING:
  uniform:
    U ~ Uniform(0,1)
  transformation:
    X := F_inverse(U)
  output:
    X ~ F

ACCEPTANCE_REJECTION:
  proposal:
    q(x)
  target:
    p(x)
  envelope:
    p(x) <= M q(x)
  flow:
    SAMPLE_FROM_q
      -> SAMPLE_UNIFORM
      -> ACCEPT_OR_REJECT
      -> REPEAT

IMPORTANCE_SAMPLING:
  target:
    I := E_p[f(X)]
  proposal:
    q(x)
  weight:
    w(x) := p(x)/q(x)
  estimator:
    I_hat := (1/N)*SUM_i f(X_i)w(X_i)
  samples:
    X_i ~ q
  purpose:
    IMPROVE_EFFICIENCY_AND_RARE_EVENT_ESTIMATION

VARIANCE_REDUCTION:
  methods:
    - ANTITHETIC_VARIATES
    - CONTROL_VARIATES
    - IMPORTANCE_SAMPLING
    - STRATIFIED_SAMPLING
    - LATIN_HYPERCUBE_SAMPLING
    - CONDITIONAL_MONTE_CARLO
    - QUASI_MONTE_CARLO

ANTITHETIC_VARIATES:
  example:
    U
    AND:
      1-U
  objective:
    REDUCE_ESTIMATOR_VARIANCE

CONTROL_VARIATES:
  estimator:
    Y_cv := Y-beta*(C-E[C])
  objective:
    REDUCE_VARIANCE_USING_CORRELATED_CONTROL

STRATIFIED_SAMPLING:
  flow:
    PARTITION_SAMPLE_SPACE
      -> SAMPLE_WITHIN_STRATA
      -> WEIGHT_STRATUM_ESTIMATES
      -> COMBINE

LATIN_HYPERCUBE_SAMPLING:
  alias:
    LHS
  purpose:
    SPACE_FILLING_UNCERTAINTY_PROPAGATION

QUASI_MONTE_CARLO:
  samples:
    LOW_DISCREPANCY_SEQUENCE
  examples:
    - SOBOL_SEQUENCE
    - HALTON_SEQUENCE
  distinction:
    QUASI_MONTE_CARLO != ORDINARY_PSEUDORANDOM_MONTE_CARLO

RARE_EVENT_SIMULATION:
  methods:
    - IMPORTANCE_SAMPLING
    - SPLITTING
    - SUBSET_SIMULATION
    - CONDITIONAL_MONTE_CARLO

UNCERTAINTY_PROPAGATION:
  uncertain_inputs:
    X_1,...,X_d
  joint_distribution:
    p(X_1,...,X_d)
  model:
    Y := g(X_1,...,X_d)
  flow:
    SAMPLE_JOINT_INPUTS
      -> EXECUTE_MODEL
      -> BUILD_OUTPUT_DISTRIBUTION
      -> COMPUTE_SUMMARIES
  outputs:
    - MEAN
    - VARIANCE
    - QUANTILES
    - EXCEEDANCE_PROBABILITIES
    - RISK_MEASURES

DEPENDENT_INPUTS:
  requirement:
    SAMPLE_FROM_JOINT_DISTRIBUTION
  methods:
    - MULTIVARIATE_DISTRIBUTION
    - COPULA
    - CONDITIONAL_SAMPLING
    - CORRELATION_TRANSFORM
  caution:
    INDEPENDENT_SAMPLING_CAN_MISREPRESENT_RISK_WHEN_INPUTS_DEPENDENT

CORRELATED_NORMAL_SAMPLING:
  covariance:
    Sigma
  decomposition:
    Sigma := L L_transpose
  independent_standard_normal:
    Z ~ N(0,I)
  transformation:
    X := mu + LZ

RISK_ANALYSIS_CONNECTION:
  random_loss:
    L
  simulation:
    GENERATES:
      L_1,...,L_N
  estimates:
    - EXPECTED_LOSS
    - VALUE_AT_RISK
    - EXPECTED_SHORTFALL
    - EXCEEDANCE_PROBABILITY
    - LOSS_DISTRIBUTION

VALUE_AT_RISK_ESTIMATION:
  estimator:
    EMPIRICAL_alpha_QUANTILE
  caution:
    TAIL_QUANTILE_PRECISION_MAY_REQUIRE_LARGE_SAMPLE

EXPECTED_SHORTFALL_ESTIMATION:
  method:
    AVERAGE_SIMULATED_LOSSES_IN_RELEVANT_UPPER_TAIL
  caution:
    FINITE_SAMPLE_TAIL_ESTIMATE_CAN_HAVE_HIGH_VARIANCE

DECISION_THEORY_CONNECTION:
  action:
    a
  uncertain_outcome:
    Y_a
  simulation:
    ESTIMATE:
      E[U(Y_a)]
      OR:
        E[L(Y_a)]
  decision:
    COMPARE_ACTIONS_USING_SIMULATED_EXPECTED_CRITERIA

EXPECTED_VALUE_CONNECTION:
  target:
    E[g(X)]
  simulation_estimator:
    SAMPLE_AVERAGE

BAYESIAN_CONNECTION:
  posterior:
    p(theta|D)
  posterior_predictive:
    p(Y_new|D)
  flow:
    SAMPLE_theta_FROM_POSTERIOR
      -> SAMPLE_OR_COMPUTE_Y_new
      -> ESTIMATE_POSTERIOR_PREDICTIVE_QUANTITIES

MARKOV_CHAIN_MONTE_CARLO_CONNECTION:
  alias:
    MCMC
  purpose:
    SAMPLE_FROM_DISTRIBUTIONS_DIFFICULT_TO_SAMPLE_DIRECTLY
  distinction:
    MCMC_SAMPLES_ARE_GENERALLY_DEPENDENT
    WHILE:
      BASIC_MONTE_CARLO_OFTEN_ASSUMES_IID_SAMPLES

MCMC_ESTIMATOR:
  estimator:
    mu_hat := (1/N)*SUM_i f(X_i)
  convergence:
    TO_TARGET_EXPECTATION
    UNDER:
      APPROPRIATE_ERGODIC_CONDITIONS

AUTOCORRELATION:
  consequence:
    EFFECTIVE_SAMPLE_SIZE_LESS_THAN_RAW_SAMPLE_COUNT
    IN_GENERAL

BURN_IN:
  purpose:
    REDUCE_INFLUENCE_OF_INITIAL_STATE
  caution:
    BURN_IN_LENGTH_HAS_NO_UNIVERSAL_FIXED_VALUE

THINNING:
  method:
    RETAIN_EVERY_kth_SAMPLE
  caution:
    OFTEN_DISCARDS_INFORMATION_AND_IS_NOT_AUTOMATICALLY_REQUIRED

CONVERGENCE_DIAGNOSTICS:
  methods:
    - TRACE_PLOTS
    - MULTIPLE_CHAINS
    - EFFECTIVE_SAMPLE_SIZE
    - R_HAT
    - AUTOCORRELATION_ANALYSIS
  caution:
    NO_SINGLE_DIAGNOSTIC_PROVES_CONVERGENCE

SIMULATION_BIAS:
  sources:
    - MODEL_MISSPECIFICATION
    - INPUT_DISTRIBUTION_ERROR
    - NUMERICAL_APPROXIMATION
    - FINITE_SAMPLE_BIAS
    - RANDOM_NUMBER_DEFECT
    - IMPLEMENTATION_ERROR
  rule:
    MORE_SIMULATIONS_REDUCE_MONTE_CARLO_SAMPLING_ERROR
    BUT:
      DO_NOT_REMOVE_MODEL_BIAS

MODEL_RISK:
  response:
    - MODEL_VALIDATION
    - SENSITIVITY_ANALYSIS
    - SCENARIO_ANALYSIS
    - STRESS_TESTING
    - ALTERNATIVE_MODEL_COMPARISON

SENSITIVITY_ANALYSIS:
  flow:
    VARY_INPUT_OR_PARAMETER
      -> RERUN_SIMULATION
      -> COMPARE_OUTPUT_DISTRIBUTIONS
      -> IDENTIFY_INFLUENTIAL_FACTORS

SCENARIO_ANALYSIS:
  simulation:
    PROPAGATE_REMAINING_UNCERTAINTY_WITHIN_SCENARIO

STRESS_TESTING:
  inputs:
    EXTREME_BUT_PLAUSIBLE_VALUES_OR_DISTRIBUTIONS
  purpose:
    ASSESS:
      TAIL_OUTCOMES
      SYSTEM_RESILIENCE
      DECISION_ROBUSTNESS

STOPPING_RULE:
  possible_criteria:
    - FIXED_SAMPLE_COUNT
    - TARGET_STANDARD_ERROR
    - TARGET_CONFIDENCE_INTERVAL_WIDTH
    - COMPUTATIONAL_BUDGET

COMPUTATIONAL_COST:
  approximate:
    COST := N*COST_PER_MODEL_RUN
  consequence:
    EXPENSIVE_MODELS_MAY_REQUIRE:
      SURROGATE_MODELS
      PARALLELISM
      VARIANCE_REDUCTION

PARALLEL_MONTE_CARLO:
  property:
    INDEPENDENT_TRIALS_CAN_OFTEN_RUN_IN_PARALLEL
  requirement:
    RANDOM_STREAMS_SHOULD_BE_MANAGED_TO_AVOID_UNINTENDED_OVERLAP

SURROGATE_MODEL:
  role:
    APPROXIMATE_EXPENSIVE_SIMULATOR
  examples:
    - GAUSSIAN_PROCESS
    - POLYNOMIAL_CHAOS
    - NEURAL_NETWORK
    - RESPONSE_SURFACE
  caution:
    SURROGATE_ERROR_MUST_BE_INCLUDED_WHEN_MATERIAL

REPRODUCIBILITY:
  preserve:
    - MODEL_VERSION
    - INPUT_DISTRIBUTIONS
    - PARAMETER_VALUES
    - RANDOM_GENERATOR
    - RANDOM_SEED
    - SAMPLE_COUNT
    - SOFTWARE_VERSION
    - NUMERICAL_PRECISION

APPLICATIONS:
  MONTE_CARLO_SIMULATION -> SUPPORTS -> RISK_ANALYSIS
  MONTE_CARLO_SIMULATION -> SUPPORTS -> DECISION_THEORY
  MONTE_CARLO_SIMULATION -> SUPPORTS -> EXPECTED_VALUE_ESTIMATION
  MONTE_CARLO_SIMULATION -> SUPPORTS -> UNCERTAINTY_QUANTIFICATION
  MONTE_CARLO_SIMULATION -> SUPPORTS -> BAYESIAN_STATISTICS
  MONTE_CARLO_SIMULATION -> SUPPORTS -> FINANCE
  MONTE_CARLO_SIMULATION -> SUPPORTS -> ENGINEERING
  MONTE_CARLO_SIMULATION -> SUPPORTS -> STOCHASTIC_MODELLING

MODEL_CONNECTIONS:
  MONTE_CARLO_SIMULATION -> CONNECTS_TO -> EXPECTED_VALUE
  MONTE_CARLO_SIMULATION -> CONNECTS_TO -> RISK_ANALYSIS
  MONTE_CARLO_SIMULATION -> CONNECTS_TO -> DECISION_THEORY
  MONTE_CARLO_SIMULATION -> CONNECTS_TO -> MCMC
  MONTE_CARLO_SIMULATION -> CONNECTS_TO -> IMPORTANCE_SAMPLING
  MONTE_CARLO_SIMULATION -> CONNECTS_TO -> RANDOM_VARIABLES
  MONTE_CARLO_SIMULATION -> CONNECTS_TO -> PROBABILITY_DISTRIBUTIONS

SEMANTIC_EQUIVALENCE:
  "Monte Carlo simulation"
    ~= "stochastic numerical estimation by repeated random sampling"

  "Monte Carlo estimator"
    ~= "sample-based stochastic estimator"

  "Monte Carlo error"
    ~= "sampling error arising from finite simulation count"

  EXCLUDE_EQUIVALENCE:
    MONTE_CARLO_SIMULATION != MCMC
    MONTE_CARLO_ERROR != MODEL_ERROR
    RANDOM_SAMPLE != DETERMINISTIC_GRID
    SAMPLE_COUNT != EFFECTIVE_SAMPLE_SIZE
    QUASI_MONTE_CARLO != ORDINARY_MONTE_CARLO
    EXPECTED_VALUE != GUARANTEED_OUTCOME

VALIDATION_RULES:
  RULE_1:
    INPUT_DISTRIBUTIONS_AND_DEPENDENCE_STRUCTURE_MUST_BE_DEFINED
  RULE_2:
    RANDOM_GENERATOR_AND_SEED_SHOULD_BE_RECORDED_WHEN_REPRODUCIBILITY_REQUIRED
  RULE_3:
    SAMPLE_COUNT_MUST_BE_SUFFICIENT_FOR_REQUIRED_PRECISION
  RULE_4:
    MONTE_CARLO_STANDARD_ERROR_MUST_ACCOUNT_FOR_SAMPLE_DEPENDENCE_WHEN_PRESENT
  RULE_5:
    IMPORTANCE_SAMPLING_SUPPORT_AND_WEIGHT_CONDITIONS_MUST_BE_VALID
  RULE_6:
    TAIL_RISK_ESTIMATES_REQUIRE_EXPLICIT_FINITE_SAMPLE_UNCERTAINTY
  RULE_7:
    MODEL_AND_INPUT_UNCERTAINTY_MUST_NOT_BE_CONFUSED_WITH_MONTE_CARLO_SAMPLING_ERROR
  RULE_8:
    CONVERGENCE_DIAGNOSTICS_MUST_MATCH_THE_SAMPLING_METHOD

CONFIDENCE:
  ontology_identity: 0.99
  expectation_estimation_structure: 0.99
  uncertainty_propagation_structure: 0.99
  variance_reduction_structure: 0.99
  risk_analysis_relationship: 0.99
  mcmc_distinction: 0.99
  semantic_distinctions: 0.99

UPDATE_MODEL:
  IF:
    SAMPLE_COUNT_INCREASES
  THEN:
    UPDATE:
      ESTIMATES
      STANDARD_ERRORS
      CONFIDENCE_INTERVALS
      TAIL_RISK_ESTIMATES

  IF:
    INPUT_DISTRIBUTIONS_CHANGE
  THEN:
    UPDATE:
      RANDOM_SAMPLING_MODEL
    RERUN:
      SIMULATION
    RECOMPUTE:
      OUTPUT_DISTRIBUTION
      RISK_MEASURES

  IF:
    MODEL_VERSION_CHANGES
  THEN:
    UPDATE:
      SIMULATION_MODEL
    RERUN:
      VALIDATION
      SENSITIVITY_ANALYSIS
      MONTE_CARLO_ESTIMATION

  PRESERVE:
    MODEL_VERSION
    INPUT_DISTRIBUTIONS
    DEPENDENCE_STRUCTURE
    RANDOM_GENERATOR
    RANDOM_SEED
    SAMPLE_COUNT
    ESTIMATOR_DEFINITION
    DATA_PROVENANCE

AI_CONSUMPTION:
  parse_as: FORMAL_SEMANTIC_SPECIFICATION
  execution_mode: RECURSIVE_REASONING

  preferred_reasoning_order:
    - IDENTIFY_TARGET_QUANTITY
    - IDENTIFY_INPUT_DISTRIBUTIONS
    - IDENTIFY_DEPENDENCE_STRUCTURE
    - SELECT_SAMPLING_METHOD
    - GENERATE_RANDOM_SAMPLES
    - EXECUTE_MODEL_FOR_EACH_SAMPLE
    - COMPUTE_MONTE_CARLO_ESTIMATOR
    - QUANTIFY_MONTE_CARLO_ERROR
    - APPLY_VARIANCE_REDUCTION_IF_REQUIRED
    - ASSESS_CONVERGENCE_AND_EFFECTIVE_SAMPLE_SIZE
    - COMPUTE_RISK_OR_DECISION_QUANTITIES
    - RETURN_SIMULATION_INTERPRETATION

  constraints:
    - PRESERVE_INPUT_DISTRIBUTIONS
    - PRESERVE_DEPENDENCE_STRUCTURE
    - DISTINGUISH_MONTE_CARLO_ERROR_FROM_MODEL_ERROR
    - DISTINGUISH_IID_MONTE_CARLO_FROM_MCMC
    - DOCUMENT_RANDOM_GENERATOR_AND_SEED
    - QUANTIFY_FINITE_SAMPLE_UNCERTAINTY
    - PRESERVE_DATA_PROVENANCE

FLOW_HASH:
  canonical_entity: "monte-carlo-simulation-ontology-2"
  namespace: "probability-statistics/risk-decision-mathematics"
  version: "4.0"
  identity_basis:
    - PAGE_URL
    - CANONICAL_ENTITY
    - ENTITY_TYPE
    - RANDOM_SAMPLING
    - MONTE_CARLO_ESTIMATOR
    - UNCERTAINTY_PROPAGATION
    - VARIANCE_REDUCTION

END_FLOW