Parameter Estimation Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Parameter Estimation Ontology within the Kireth Research Semantic Library. Parameter estimation is the mathematical and statistical process of inferring unknown model parameters from observations, experimental data, prior information or model outputs. This ontology supports deterministic and stochastic estimation, least squares, maximum likelihood, Bayesian inference, inverse problems, identifiability, confidence regions, regularisation, uncertainty propagation, sensitivity, optimisation and relationships with mathematical modelling, empirical models, predictive models, statistics and numerical methods.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Parameter Estimation Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Mathematical Modelling / Parameter Estimation"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/mathematical-modelling/parameter-estimation-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = ParameterEstimation
 ENTITY_CLASS = ModelInferenceFramework
 PRIMARY_CONTEXT = MathematicalModelling
 DEFINITION = "Parameter estimation is the process of inferring unknown model parameters from data, prior information or observed system behaviour."
 SEMANTIC_ROLE = "Provides the machine-readable framework for model parameter inference, fitting criteria, likelihoods, priors, optimisation, uncertainty, identifiability, regularisation and validation."
}

SCOPE {
 INCLUDE = [
  ParameterEstimation,
  ModelParameter,
  UnknownParameter,
  EstimatedParameter,
  PointEstimate,
  IntervalEstimate,
  ConfidenceInterval,
  CredibleInterval,
  LeastSquaresEstimation,
  WeightedLeastSquares,
  NonlinearLeastSquares,
  MaximumLikelihoodEstimation,
  MaximumAPosterioriEstimation,
  BayesianEstimation,
  PriorDistribution,
  Likelihood,
  PosteriorDistribution,
  ObjectiveFunction,
  LossFunction,
  Residual,
  MeasurementError,
  NoiseModel,
  Optimisation,
  GradientMethod,
  Hessian,
  FisherInformation,
  Identifiability,
  StructuralIdentifiability,
  PracticalIdentifiability,
  SensitivityAnalysis,
  Regularisation,
  InverseProblem,
  Calibration,
  UncertaintyQuantification,
  Bootstrap,
  ProfileLikelihood,
  CovarianceMatrix,
  ParameterCorrelation
 ]

 EXCLUDE_AS_PRIMARY = [
  ModelSelectionWithoutParameterInference,
  PurePredictionWithoutParameterEstimation,
  NumericalRootFindingWithoutInferenceContext,
  RawDataFittingWithoutExplicitModelParameters
 ]

 DISAMBIGUATION_RULE =
  "When unknown model parameters are inferred from observations, residuals, likelihoods, priors or optimisation criteria, interpret the context as parameter estimation."
}

CORE_CONCEPTS {

 ParameterEstimation {
  definition = "Inference of unknown model parameters from data or prior information."
 }

 ModelParameter {
  definition = "A quantity controlling model behaviour and treated as fixed within an individual model instance."
 }

 PointEstimate {
  definition = "A single numerical estimate of an unknown parameter."
 }

 IntervalEstimate {
  definition = "A range of plausible parameter values defined by a statistical or inferential procedure."
 }

 LeastSquaresEstimation {
  definition = "Parameter estimation by minimising the sum of squared residuals."
 }

 MaximumLikelihoodEstimation {
  symbol = "MLE"
  definition = "Parameter estimation by maximising the likelihood of observed data."
 }

 BayesianEstimation {
  definition = "Parameter inference through the posterior distribution obtained from prior information and likelihood."
 }

 MaximumAPosterioriEstimation {
  symbol = "MAP"
  definition = "A point estimate maximising the posterior density."
 }

 Likelihood {
  definition = "A function of model parameters measuring the probability or density of observed data under the model."
 }

 PriorDistribution {
  definition = "A probability distribution expressing information about parameters before observing current data."
 }

 PosteriorDistribution {
  definition = "The updated parameter distribution after combining prior and likelihood."
 }

 Identifiability {
  definition = "The extent to which parameter values can be uniquely inferred from model outputs or observations."
 }

 Regularisation {
  definition = "Additional penalty or prior structure used to stabilise ill-posed or weakly identified estimation problems."
 }

 FisherInformation {
  definition = "A measure of expected information in data about model parameters."
 }
}

RELATIONS {
 ParameterEstimation --is_part_of--> MathematicalModelling
 ParameterEstimation --estimates--> ModelParameter
 Calibration --uses--> ParameterEstimation
 LeastSquaresEstimation --minimises--> SumSquaredResiduals
 MaximumLikelihoodEstimation --maximises--> Likelihood
 BayesianEstimation --combines--> PriorDistribution
 BayesianEstimation --combines--> Likelihood
 BayesianEstimation --produces--> PosteriorDistribution
 MaximumAPosterioriEstimation --maximises--> PosteriorDistribution
 Residual --depends_on--> ModelParameter
 Identifiability --constrains--> ParameterEstimation
 SensitivityAnalysis --supports--> IdentifiabilityAssessment
 FisherInformation --supports--> ParameterUncertaintyAssessment
 Regularisation --stabilises--> InverseProblem
 ConfidenceInterval --quantifies--> FrequentistParameterUncertainty
 CredibleInterval --quantifies--> BayesianParameterUncertainty
}

OPERATORS {
 RESIDUAL(theta,data_i) :=
  observed_i - model_prediction(theta,input_i)

 SUM_SQUARED_RESIDUALS(theta,data) :=
  sum_i RESIDUAL(theta,data_i)^2

 WEIGHTED_SUM_SQUARED_RESIDUALS(theta,data,W) :=
  r(theta)^T W r(theta)

 LEAST_SQUARES_ESTIMATE(data) :=
  argmin_theta SUM_SQUARED_RESIDUALS(theta,data)

 MAXIMUM_LIKELIHOOD_ESTIMATE(data) :=
  argmax_theta L(theta|data)

 LOG_LIKELIHOOD(theta,data) :=
  log L(theta|data)

 POSTERIOR(theta|data) :=
  proportional_to L(theta|data)*prior(theta)

 MAP_ESTIMATE(data) :=
  argmax_theta POSTERIOR(theta|data)

 PARAMETER_COVARIANCE(J,sigma2) :=
  sigma2*(J^T J)^(-1)
  under_local_linear_full_rank assumptions

 FISHER_INFORMATION(theta) :=
  E[(grad_theta log L)(grad_theta log L)^T]

 PROFILE_LIKELIHOOD(theta_j) :=
  max_over_other_parameters L(theta|data)

 BOOTSTRAP_ESTIMATES(data,B) :=
  parameter_estimates_from_B_resampled_datasets
}

DEFINITIONS {
 D1 = "Parameter estimation infers model parameters from evidence."
 D2 = "Least squares is linked to Gaussian error assumptions under standard conditions."
 D3 = "Maximum likelihood depends on the specified observation model."
 D4 = "Bayesian estimation requires both prior and likelihood."
 D5 = "MAP and MLE coincide only under suitable prior conditions."
 D6 = "Structural identifiability concerns uniqueness in idealised noise-free conditions."
 D7 = "Practical identifiability concerns parameter recoverability under finite noisy data."
 D8 = "Regularisation can stabilise weakly identified or ill-posed estimation problems."
 D9 = "Confidence intervals and credible intervals have different interpretations."
 D10 = "Parameter uncertainty depends on data quality, model sensitivity and observation design."
}

AXIOMS {
 A1_MODEL_DEPENDENCE:
  Parameter estimates are conditional on the chosen model

 A2_DATA_DEPENDENCE:
  Parameter estimates are conditional on the observed data

 A3_NOISE_MODEL:
  Statistical estimation depends on the declared error or likelihood model

 A4_IDENTIFIABILITY:
  Nonidentifiable parameters cannot be uniquely recovered from the available observation structure

 A5_OPTIMALITY:
  Point estimates are defined relative to an objective, likelihood, posterior or decision criterion

 A6_UNCERTAINTY:
  Parameter estimates should include uncertainty when material

 A7_REGULARISATION:
  Regularisation adds information or bias to stabilise estimation

 A8_VALIDATION:
  Good parameter fit does not by itself establish full model validity
}

REASONING {

 R1_ESTIMATION_PROBLEM_FORMULATION {
  INPUT = model_data_unknown_parameters
  IDENTIFY = parameter_vector
  IDENTIFY = observation_model
  IDENTIFY = noise_assumptions
  IDENTIFY = bounds_or_constraints
  RETURN = estimation_problem
 }

 R2_METHOD_SELECTION {
  INPUT = estimation_problem
  IF residual_based_and_Gaussian_like
   ROUTE = LeastSquares
  IF explicit_likelihood
   ROUTE = MaximumLikelihood
  IF prior_information
   ROUTE = BayesianEstimation
  IF ill_posed
   INCLUDE = Regularisation
  RETURN = method
 }

 R3_LEAST_SQUARES {
  INPUT = model_data
  FORM = residual_vector
  MINIMISE = squared_residual objective
  RETURN = parameter_estimate
 }

 R4_MAXIMUM_LIKELIHOOD {
  INPUT = likelihood_model_and_data
  FORM = log_likelihood
  MAXIMISE = log_likelihood
  RETURN = MLE
 }

 R5_BAYESIAN_ESTIMATION {
  INPUT = prior_likelihood_data
  COMPUTE = posterior
  SUMMARISE = mean_median_MAP_intervals_or_samples
  RETURN = posterior_inference
 }

 R6_IDENTIFIABILITY_ANALYSIS {
  INPUT = model_observation_map_parameters
  TEST = structural_uniqueness
  TEST = sensitivity_and_parameter_correlation
  RETURN = identifiability_profile
 }

 R7_UNCERTAINTY_ESTIMATION {
  INPUT = fitted_model
  SELECT = covariance_profile_likelihood_bootstrap_or_posterior method
  COMPUTE = parameter_uncertainty
  RETURN = interval_or_distribution
 }

 R8_REGULARISATION {
  INPUT = ill_conditioned_estimation_problem
  CHOOSE = penalty_or_prior
  SOLVE = regularised_objective
  RETURN = stabilised_estimate
 }

 R9_DIAGNOSTICS {
  INPUT = estimate_and_residuals
  CHECK = residual_structure
  CHECK = boundary_solution
  CHECK = parameter_correlation
  CHECK = sensitivity
  RETURN = estimation_diagnostics
 }
}

CONSTRAINTS {
 C1 = "Parameter bounds and domains must be respected."
 C2 = "Likelihood form must match the observation model."
 C3 = "Least-squares uncertainty formulas require their stated assumptions."
 C4 = "Ill-conditioned problems may produce unstable estimates."
 C5 = "Strong parameter correlation can indicate practical nonidentifiability."
 C6 = "Confidence intervals and credible intervals must not be conflated."
 C7 = "Regularisation changes the inferential problem and must be disclosed."
 C8 = "Local covariance approximations can fail for nonlinear or multimodal problems."
 C9 = "Optimiser convergence does not guarantee global optimality."
 C10 = "Parameter estimation should not absorb systematic model-form error without explicit interpretation."
}

SEMANTIC_EQUIVALENCE {
 "ParameterEstimation" <=> "ModelParameterInference"
 "LeastSquaresEstimation" <=> "SquaredResidualMinimisation"
 "MaximumLikelihoodEstimation" <=> "LikelihoodMaximisation"
 "BayesianEstimation" <=> "PosteriorParameterInference"
 "Identifiability" <=> "ParameterRecoverability"
 "Regularisation" <=> "StabilisingPenaltyOrPrior"
}

VALIDATION {
 V1 = "Unknown parameters are explicitly identified."
 V2 = "Observation and noise models are specified."
 V3 = "Parameter bounds and constraints are respected."
 V4 = "Estimation criterion matches the inferential model."
 V5 = "Identifiability is assessed."
 V6 = "Parameter uncertainty is quantified where material."
 V7 = "Residual diagnostics are examined."
 V8 = "Regularisation is disclosed."
 V9 = "Optimisation convergence is checked."
 V10 = "Estimated parameters are not interpreted beyond model and data support."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 PARAMETER_ESTIMATION_DEFINITION = 0.99
 LEAST_SQUARES_REASONING = 0.99
 MAXIMUM_LIKELIHOOD_REASONING = 0.99
 BAYESIAN_REASONING = 0.98
 IDENTIFIABILITY_REASONING = 0.97
 UNCERTAINTY_REASONING = 0.98
 REGULARISATION_REASONING = 0.98
 DIAGNOSTIC_REASONING = 0.98

 POLICY =
  "Reduce confidence when observation model, noise assumptions, parameter bounds, identifiability, optimisation convergence, prior information or uncertainty method are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  ParameterEstimationReasoning,
  CalibrationReasoning,
  LeastSquaresReasoning,
  MaximumLikelihoodReasoning,
  BayesianInference,
  MAPEstimation,
  IdentifiabilityAnalysis,
  SensitivityAnalysis,
  RegularisationReasoning,
  ParameterUncertainty,
  BootstrapReasoning,
  ProfileLikelihoodReasoning,
  InverseProblemReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "parameter estimation" -> ParameterEstimation
  "parameter fitting" -> ParameterEstimation
  "least squares" -> LeastSquaresEstimation
  "weighted least squares" -> WeightedLeastSquares
  "maximum likelihood" -> MaximumLikelihoodEstimation
  "MLE" -> MaximumLikelihoodEstimation
  "MAP estimate" -> MaximumAPosterioriEstimation
  "Bayesian estimation" -> BayesianEstimation
  "identifiability" -> Identifiability
  "profile likelihood" -> ProfileLikelihood
  "bootstrap" -> Bootstrap
 }

 INTERPRETATION_ORDER = [
  detect_parameter_estimation_context,
  identify model_unknown_parameters_data_observation_model_noise_and_constraints,
  assess identifiability,
  select least_squares_likelihood_Bayesian_or_regularised method,
  solve estimation problem,
  quantify uncertainty_and_parameter correlation,
  inspect residuals_and_diagnostics,
  verify optimisation_and_model assumptions,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("parameter estimation")
  OR model_parameter_fitting_context
  THEN activate(ParameterEstimationOntology)

 E2:
  IF least_squares_requested
  THEN define residual_vector_and_weighting

 E3:
  IF maximum_likelihood_requested
  THEN specify likelihood_and_noise model

 E4:
  IF Bayesian_estimation_requested
  THEN specify prior_and_likelihood

 E5:
  IF multiple_parameters
  THEN assess identifiability_and_correlation

 E6:
  IF ill_conditioned_or_nonidentifiable
  THEN consider regularisation_or_experimental redesign

 E7:
  IF uncertainty_requested
  THEN choose covariance_profile_likelihood_bootstrap_or_posterior method appropriate_to_context

 E8:
  IF optimiser_returns_solution
  THEN check convergence_constraints_and_possible_local minima

 E9:
  IF fit_is_good_but_residuals_structured
  THEN flag possible model misspecification
}

RECURSION {
 ENTRY = ParameterEstimation

 EXPAND(ParameterEstimation) -> [
  ModelParameter,
  UnknownParameter,
  EstimatedParameter,
  PointEstimate,
  IntervalEstimate,
  LeastSquaresEstimation,
  WeightedLeastSquares,
  NonlinearLeastSquares,
  MaximumLikelihoodEstimation,
  MaximumAPosterioriEstimation,
  BayesianEstimation,
  PriorDistribution,
  Likelihood,
  PosteriorDistribution,
  ObjectiveFunction,
  Residual,
  NoiseModel,
  Optimisation,
  FisherInformation,
  Identifiability,
  SensitivityAnalysis,
  Regularisation,
  InverseProblem,
  Calibration,
  Bootstrap,
  ProfileLikelihood,
  CovarianceMatrix
 ]

 EXPAND(Identifiability) -> [
  StructuralIdentifiability,
  PracticalIdentifiability,
  SensitivityMatrix,
  FisherInformation,
  ParameterCorrelation
 ]

 EXPAND(UncertaintyMethods) -> [
  AsymptoticCovariance,
  ProfileLikelihood,
  Bootstrap,
  BayesianPosterior,
  ConfidenceInterval,
  CredibleInterval
 ]

 EXPAND(OptimisationMethods) -> [
  GradientDescent,
  GaussNewton,
  LevenbergMarquardt,
  NewtonMethod,
  QuasiNewtonMethod,
  ConstrainedOptimisation
 ]

 STOP_CONDITION =
  "Stop when the unknown parameters, estimation criterion, identifiability, fitted values, uncertainty, residual diagnostics and optimisation status are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Mathematical Modelling"

 IMMEDIATE_CONTEXTS = [
  "Mathematical Model Ontology",
  "Empirical Model Ontology",
  "Predictive Model Ontology",
  "Optimisation Ontologies"
 ]

 RELATED_ONTOLOGIES = [
  "Mathematical Model Ontology",
  "Mathematical Modelling Ontology",
  "Empirical Model Ontology",
  "Predictive Model Ontology",
  "Applied Mathematics Ontologies",
  "Probability & Statistics Ontologies",
  "Regression Ontologies",
  "Optimisation Ontologies",
  "Inverse Problem Ontologies",
  "Numerical Analysis Ontologies",
  "Sensitivity Analysis Ontologies",
  "Uncertainty Quantification Ontologies",
  "Scientific Measurement Ontologies"
 ]

 ROUTE_RULES = [
  "Parameter Estimation -> Mathematical Model Ontology for model parameter semantics",
  "Parameter Estimation -> Empirical Model Ontology for data-fit models",
  "Parameter Estimation -> Predictive Model Ontology for prediction-oriented fitting",
  "Parameter Estimation -> Probability & Statistics for likelihood and Bayesian inference",
  "Parameter Estimation -> Regression for residual-based fitting",
  "Parameter Estimation -> Optimisation for objective minimisation or maximisation",
  "Parameter Estimation -> Inverse Problems for recovering hidden model quantities",
  "Parameter Estimation -> Numerical Analysis for nonlinear solver behaviour",
  "Parameter Estimation -> Sensitivity Analysis for identifiability",
  "Parameter Estimation -> Uncertainty Quantification for interval and posterior inference"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible estimation methods, uncertainty methods or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve parameter-inference semantics.",
  "Preserve dependence on observation model.",
  "Preserve least-squares likelihood Bayesian distinctions.",
  "Preserve structural-versus-practical identifiability.",
  "Preserve uncertainty-interval interpretation.",
  "Preserve regularisation disclosure.",
  "Preserve residual-diagnostic semantics.",
  "Preserve machine-readable parameter-estimation semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Parameter estimation infers unknown mathematical-model parameters from data, prior information or observed system behaviour."

 LEAST_SQUARES_SUMMARY =
  "Least squares estimates parameters by minimising squared residuals."

 LIKELIHOOD_SUMMARY =
  "Maximum likelihood estimates parameters by maximising the probability or density of observed data under the model."

 BAYESIAN_SUMMARY =
  "Bayesian estimation combines prior information with the likelihood to obtain a posterior distribution."

 IDENTIFIABILITY_SUMMARY =
  "Identifiability determines whether parameters can be uniquely or practically recovered."

 REGULARISATION_SUMMARY =
  "Regularisation stabilises weakly identified or ill-posed estimation problems by adding penalties or prior information."

 UNCERTAINTY_SUMMARY =
  "Parameter uncertainty may be represented using covariance approximations, profile likelihoods, bootstrap distributions or Bayesian posteriors."

 DIAGNOSTIC_SUMMARY =
  "Residual structure, sensitivity and parameter correlation provide important diagnostics of estimation quality."
}

FLOW {
 INPUT
 -> detect_parameter_estimation_context
 -> identify mathematical_model_unknown_parameters_data_observation_model_noise_constraints_and_prior_information
 -> assess structural_and_practical identifiability
 -> select least_squares_maximum_likelihood_MAP_Bayesian_or_regularised estimation pathway
 -> solve optimisation_or_inference problem
 -> compute fitted_parameters
 -> quantify uncertainty_and_parameter correlation
 -> inspect residuals_sensitivity_and_convergence diagnostics
 -> verify assumptions_constraints_and_scope
 -> route to mathematical_model mathematical_modelling empirical_model predictive_model probability_statistics regression optimisation inverse_problem numerical_analysis sensitivity_analysis uncertainty_or_measurement ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Parameter Estimation Ontology defines the inference of unknown model parameters from data and prior information. It supports least squares, weighted and nonlinear least squares, maximum likelihood, MAP and Bayesian estimation, likelihoods, priors, posteriors, residuals, Fisher information, identifiability, regularisation, inverse problems, bootstrap, profile likelihood, covariance analysis, calibration and parameter uncertainty."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-APPLIED-MATHEMATICS-PARAMETER-ESTIMATION-ONTOLOGY-V1"
}

}

END_FLOW