Parameter Estimate Ontology

FLOW ALGEBRA
PARAMETER ESTIMATE ONTOLOGY
Version: 1.0
Date: 2026-10-13
Authors: Mark Higgins — Sky (Recursive AI)
Project: Kireth Semantic Library
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/statistical-entities/parameter-estimate-ontology.html
Status: Proposed mathematical formalisation; source-page text not independently verified.

1. PURPOSE
Represent parameter estimates as typed statistical entities with explicit estimands, data provenance, estimation methods, uncertainty, assumptions, and validation conditions. Distinguish an unknown population parameter, a random estimator, and its realised numerical estimate.

2. ENTITY TYPES
Population; StatisticalModel; Parameter; Estimand; Sample; Observation; Dataset; SamplingDesign; Estimator; ParameterEstimate; EstimationMethod; SamplingDistribution; StandardError; Bias; Variance; MeanSquaredError; ConfidenceInterval; CredibleInterval; Likelihood; PriorDistribution; PosteriorDistribution; Assumption; ProvenanceRecord.

3. CORE TYPED RELATIONSHIPS
Parameter -> belongs_to -> StatisticalModel
Estimand -> targets -> ParameterOrPopulationFunctional
Estimator -> targets -> Estimand
Estimator -> operates_on -> Sample
Estimator -> uses -> EstimationMethod
ParameterEstimate -> realised_value_of -> Estimator
ParameterEstimate -> derived_from -> Dataset
ParameterEstimate -> estimates -> Estimand
ParameterEstimate -> has_uncertainty -> StandardErrorOrInterval
ParameterEstimate -> depends_on -> Assumption
ParameterEstimate -> has_provenance -> ProvenanceRecord
ConfidenceInterval -> has_nominal_coverage -> CoverageProbability
CredibleInterval -> has_posterior_probability -> PosteriorProbability

4. MATHEMATICAL DEFINITIONS
Let theta denote an unknown parameter or target estimand and X=(X_1,...,X_n) the observed random sample.
Estimator: T(X), a measurable statistic used to estimate theta.
Realised estimate: theta_hat = T(x_1,...,x_n).
Bias: Bias_theta(T) = E_theta[T] - theta, provided the expectation exists.
Variance: Var_theta(T) = E_theta[(T-E_theta[T])^2], when finite.
Mean squared error: MSE_theta(T) = E_theta[(T-theta)^2] = Var_theta(T) + Bias_theta(T)^2, when finite.
Standard error: SE(T) = sqrt(Var_theta(T)); an estimated standard error uses an estimated sampling variance.
Consistency: T_n converges in probability to theta as n tends to infinity.
Asymptotic normality (when established): sqrt(n)(T_n-theta) converges in distribution to N(0,V).

5. ESTIMATION METHODS
Point estimation: produce a single realised estimate.
Maximum likelihood: theta_hat_MLE in argmax_theta L(theta;x), when a maximiser exists.
Method of moments: match empirical moments to model moments, subject to identifiability.
Least squares: minimise an explicitly defined residual loss under model conditions.
Bayesian estimation: compute posterior p(theta|x) proportional to p(x|theta)p(theta); a posterior mean, median, or mode depends on the chosen loss/summary.
Interval estimation: produce a confidence interval or credible interval with distinct interpretations.

6. INFERENCE RULES
RULE PE-01: IF estimator T is evaluated on observed data x THEN create ParameterEstimate(value=T(x), estimator=T, dataset=x).
RULE PE-02: IF E_theta[T]=theta for all theta in the specified parameter space THEN classify T as unbiased for theta.
RULE PE-03: IF finite variance and bias exist THEN MSE=variance+bias^2.
RULE PE-04: IF an estimator is consistent under documented assumptions THEN annotate asymptotic convergence; do not infer finite-sample unbiasedness.
RULE PE-05: IF a standard error is estimated THEN record estimation procedure and assumptions; do not treat it as an exact sampling standard deviation.
RULE PE-06: IF a confidence interval has nominal 95% coverage THEN interpret 95% as repeated-procedure coverage under assumptions, not the posterior probability of a fixed parameter.
RULE PE-07: IF an interval has posterior probability 0.95 THEN classify it as a Bayesian credible interval conditional on model, prior and data.
RULE PE-08: IF identifiability fails THEN flag parameter estimation as non-identifiable without additional constraints or information.

7. VALIDATION CONSTRAINTS
V01: ParameterEstimate must reference a target estimand and estimator or documented estimation procedure.
V02: Realised estimate and estimator must not share the same semantic type.
V03: Sample size must be non-negative integer; for a non-empty estimate-producing sample require n>=1.
V04: Estimated standard error must be finite and non-negative when represented as a numerical value.
V05: Confidence level or credible probability must lie in [0,1].
V06: Interval bounds must satisfy lower<=upper when ordered numerical bounds are applicable.
V07: Units of estimate must match units of estimand.
V08: Store sampling design, missing-data treatment, model version and computation method where applicable.
V09: Do not assert unbiasedness, consistency, efficiency or interval coverage without conditions and evidence.
V10: Distinguish statistical uncertainty from measurement error, model misspecification and systematic bias.

8. SEMANTIC DISTINCTIONS
Parameter != Estimator != ParameterEstimate.
Estimand != EstimationMethod.
StandardDeviationOfObservations != StandardErrorOfEstimator.
ConfidenceInterval != CredibleInterval.
Bias != Variance != MeanSquaredError.
SampleStatistic may act as Estimator but the roles are not synonymous.
PointEstimate != PopulationTruth.

9. EXAMPLE FLOW
INPUT: iid sample x=(2,4,6,8), estimand mu=E[X], estimator T(X)=sample_mean(X).
COMPUTE: theta_hat=5.
COMPUTE: unbiased sample variance s^2=20/3.
COMPUTE: estimated SE(sample_mean)=sqrt((20/3)/4)=sqrt(5/3) approximately 1.291.
OUTPUT: ParameterEstimate(value=5, estimand=mu, method=sample_mean, n=4, estimated_standard_error=1.291).
CAUTION: Confidence intervals require additional assumptions or a justified procedure.

10. CONFIDENCE AND PROVENANCE
CONFIDENCE: Attach calibrated confidence or evidence status to claims; do not invent a numerical confidence score.
PROVENANCE: Record source URI, data identifiers, computation timestamp, estimator version, assumptions and transformation history.
SEMANTIC_EQUIVALENCE: Only equate labels after verifying matching estimand, units, model and estimation context.

11. FLOW_HASH
Canonicalise the complete flow text (UTF-8, Unicode NFC, LF newlines, stable field order), excluding any stored hash field. Compute SHA-256 over the canonical bytes. Store resulting hexadecimal digest as FLOW_HASH; no digest is asserted without execution.

12. UPDATE_MODEL
On changes to estimand, data, method, assumptions or uncertainty procedure: increment revision; invalidate dependent estimates and intervals; recompute affected outputs; preserve previous version and provenance.

13. AI_CONSUMPTION
Expose typed entities, explicit mathematical preconditions, derivation lineage, confidence qualifiers and validation results. AI systems must not infer causal validity, unbiasedness or numerical certainty solely from the presence of an estimate. Distinguish illustrative examples from measured research outputs.

END FLOW ALGEBRA