Hypothesis Ontology

FLOW ALGEBRA
HYPOTHESIS ONTOLOGY
Kireth Semantic Library | Statistical Entities
Version: 1.0 | Date: 2026-10-13
Authors: Mark Higgins — Sky (Recursive AI)
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/statistical-entities/hypothesis-ontology.html
Status: Proposed formalisation; source-page wording not independently verified.

1. PURPOSE
Represent statistical hypotheses as typed, testable claims about population parameters, distributions, or data-generating processes, with explicit assumptions, decision procedures, evidence and uncertainty.

2. ENTITY TYPES
Hypothesis := {id, proposition, target_population, model, parameter_space, parameter_constraint, role, status, provenance}
NullHypothesis <: Hypothesis [role=H0]
AlternativeHypothesis <: Hypothesis [role=H1]
SimpleHypothesis <: Hypothesis [specifies_one_distribution]
CompositeHypothesis <: Hypothesis [specifies_set_of_distributions]
DirectionalHypothesis <: Hypothesis [one_sided_constraint]
NonDirectionalHypothesis <: Hypothesis [two_sided_constraint]
StatisticalTest := {id, test_statistic, reference_distribution, rejection_region, alpha, assumptions}
TestResult := {observed_statistic, p_value?, decision, effect_size?, interval?, diagnostics}

3. FORMAL FOUNDATIONS
Let (Omega,F,P_theta) be a statistical model indexed by theta in Theta.
H0: theta in Theta_0; H1: theta in Theta_1.
For conventional exhaustive disjoint testing, Theta_0 intersect Theta_1 = empty and Theta_0 union Theta_1 = Theta.
Example: H0: mu = mu0; H1: mu != mu0.
A hypothesis is a proposition about a model, not an observed outcome or a probability assigned to that proposition by a frequentist p-value.

4. TESTING OPERATIONS
flow "EvaluateHypothesis" v1.0
  input: hypothesis_pair, data, statistical_model, alpha
  require: 0 < alpha < 1
  validate: hypotheses_well_formed; test_assumptions_checked; data_provenance_known
  derive: statistic = T(data)
  derive: p_value = P_H0(T at_least_as_extreme_as observed_T) [valid reference distribution]
  decide: reject_H0 if p_value <= alpha else fail_to_reject_H0
  output: TestResult with assumptions, uncertainty, limitations, provenance
end

5. CORE RELATIONSHIPS
Hypothesis --CONCERNS--> PopulationParameter
Hypothesis --CONSTRAINS--> ParameterSpace
Hypothesis --ASSESSED_BY--> StatisticalTest
StatisticalTest --USES--> TestStatistic
StatisticalTest --ASSUMES--> ModelAssumption
StatisticalTest --PRODUCES--> TestResult
TestResult --DERIVED_FROM--> Dataset
TestResult --REPORTS--> PValue
TestResult --MAY_REPORT--> ConfidenceInterval
TestResult --MAY_REPORT--> EffectSize

6. INFERENCE RULES
R1: valid_test AND p_value <= alpha => reject_null_at_alpha.
R2: valid_test AND p_value > alpha => fail_to_reject_null_at_alpha; NOT accept_null_as_proven.
R3: assumption_violation => qualify_or_invalidate_nominal_p_value.
R4: multiple_tests_without_adjustment => flag_familywise_or_false_discovery_risk.
R5: small_p_value DOES_NOT_IMPLY large_effect_size.
R6: p_value DOES_NOT_EQUAL probability_null_is_true.
R7: statistical_significance DOES_NOT_IMPLY causation or practical significance.
R8: preregistered_hypothesis != post_hoc_exploratory_hypothesis; preserve distinction.

7. ERROR AND POWER SEMANTICS
Type I error: reject H0 when H0 is true; nominal probability bounded by alpha under valid test design.
Type II error: fail to reject H0 when H1 is true; probability beta(theta), dependent on alternative.
Power(theta) = 1 - beta(theta) for theta in Theta_1.
A test's significance level and power are properties of the procedure under specified model conditions, not probabilities that a particular conclusion is correct.

8. SEMANTIC DISTINCTIONS
StatisticalHypothesis != ScientificTheory.
NullHypothesis != AlternativeHypothesis.
TestStatistic != PValue.
FailureToReject != ProofOfNull.
ConfidenceInterval != PosteriorCredibleInterval.
StatisticalSignificance != PracticalSignificance.
Association != Causation.

9. VALIDATION
V1: hypothesis statements must be logically interpretable within the declared model.
V2: all parameter constraints must use compatible types and units.
V3: alpha must lie strictly between zero and one.
V4: any reported p-value must lie in [0,1] and identify its null reference model.
V5: conclusions must preserve the reject/fail-to-reject distinction.
V6: disclose sampling design, test assumptions, missingness, multiplicity and data provenance.
V7: report uncertainty and effect magnitude where appropriate.

10. CONFIDENCE AND PROVENANCE
CONFIDENCE: {mathematical_definition: high, empirical_claim: evidence_dependent, source_alignment: unverified}
PROVENANCE: {source_url, dataset_id?, model_id?, method_id?, created_at, authors}
SEMANTIC_EQUIVALENCE: only if propositions, parameter constraints and model context coincide.

11. FLOW HASH AND UPDATE MODEL
FLOW_HASH: SHA-256(canonical_UTF8(normalised_entity_types, axioms, relationships, rules, validation, version))
Note: this specifies a deterministic hash construction; no hash digest is asserted here.
UPDATE_MODEL: semantic_versioning; preserve prior versions; invalidate dependent inferences when assumptions or hypotheses change.

12. AI CONSUMPTION
AI_CONSUMPTION:
  parse_as: typed_statistical_ontology
  distinguish: hypothesis, test, evidence, decision, uncertainty
  prohibit: interpreting p_value as P(H0|data)
  require: explicit assumptions and provenance for statistical conclusions
  output: machine_readable_relations_and_human_readable_explanations

END FLOW ALGEBRA