Probability Domain Ontology

FLOW ALGEBRA
TITLE: Probability Domain Ontology
DOMAIN: Domain Ontologies / Mathematical Domains / Probability
SOURCE: https://kirethresearch.org/booksai-core/ontologies/domain-ontologies/mathematical-domains/probability-domain-ontology.html
VERSION: 1.0
AUTHORS: Mark Anthony Higgins — Sky (Recursive AI)
STATUS: Companion formalisation; source-page specification not independently verified.

flow "ProbabilityDomainOntology" v1.0 {
  ontology:
    id: "probability-domain-ontology"
    parent: "mathematical-domains"
    purpose: "Represent probability spaces, random variables, distributions, events, conditioning, independence and uncertainty with explicit mathematical assumptions."

  types:
    ProbabilitySpace: {id: string, sample_space: string, sigma_algebra: string, probability_measure: string, provenance: string?}
    Event: {id: string, space_id: string, definition: string}
    RandomVariable: {id: string, space_id: string, codomain: string, measurable: boolean, unit: string?}
    Distribution: {id: string, family: string, parameters: map, support: string, normalised: boolean}
    Observation: {id: string, variable_id: string, value: number|string, timestamp: datetime?, source: string}
    ProbabilityAssessment: {id: string, event_id: string, value: number, conditions: string[], method: string, evidence: string[]}
    Relationship: {id: string, subject: string, predicate: string, object: string, assumptions: string[], confidence: number?, provenance: string[]}

  predicates:
    belongs_to(event: Event, space: ProbabilitySpace)
    defined_on(variable: RandomVariable, space: ProbabilitySpace)
    follows_distribution(variable: RandomVariable, distribution: Distribution)
    subset_of(a: Event, b: Event)
    independent_of(a: Event, b: Event)
    conditioned_on(assessment: ProbabilityAssessment, event: Event)
    observed_as(variable: RandomVariable, observation: Observation)
    derived_from(assessment: ProbabilityAssessment, evidence: string)

  constraints:
    - "Every probability value must lie within [0,1]."
    - "P(empty set)=0 and P(sample space)=1; probabilities are countably additive over disjoint events."
    - "Conditional probability P(A|B) is defined only when P(B)>0 under the elementary ratio definition."
    - "Independence is not implied by non-correlation, except under additional justified assumptions."
    - "Distinguish probability assessments from epistemic confidence in an assertion."
    - "Random variables must be measurable and distributions must be normalised."
    - "Never infer causation from probabilistic association alone."

  inference:
    rule complement:
      if probability(A)=p and same_space(A,not_A)
      then probability(not_A)=1-p
      requires: "A is a measurable event"
    rule disjoint_union:
      if disjoint(A,B) and probability(A)=p and probability(B)=q
      then probability(union(A,B))=p+q
    rule conditional_product:
      if probability(B)=q and conditional_probability(A,B)=r and q>0
      then probability(intersection(A,B))=q*r
    rule bayes:
      if conditional_probability(B,A)=likelihood and probability(A)=prior and probability(B)=evidence and evidence>0
      then conditional_probability(A,B)=likelihood*prior/evidence
    rule independence_symmetry:
      if independent_of(A,B)
      then independent_of(B,A)
      note: "Independence is symmetric, but not generally transitive."

  SEMANTIC_EQUIVALENCE:
    - "Normalise probability notation without merging distinct conditioning contexts."
    - "Distinguish independent events, independent random variables and conditional independence."
    - "Preserve sample spaces, units, assumptions and parameterisations."

  CONFIDENCE:
    range: [0,1]
    factors: [data_quality, sampling_design, model_fit, calibration, uncertainty, replication]
    policy: "Record evidence quality separately from numerical event probability; do not manufacture certainty through inference chains."

  VALIDATION:
    - "Check probability axioms, distribution support and normalisation."
    - "Verify conditioning denominators and compatible probability spaces."
    - "Check measurable events, units, parameter constraints and provenance."
    - "Flag contradictory assessments and unsupported independence assumptions."

  FLOW_HASH:
    algorithm: "SHA-256"
    input: "canonicalised ontology version, typed assertions, rules, assumptions and provenance"
    note: "Compute hash at execution; no hash value is claimed here."

  UPDATE_MODEL:
    strategy: "versioned, provenance-preserving incremental update"
    steps: [ingest, normalise, validate, compare, reconcile, recompute_inferences, publish]
    rollback: true

  AI_CONSUMPTION:
    representation: "typed probabilistic knowledge graph with explicit conditioning and assumptions"
    outputs: [event_links, distribution_descriptions, conditional_probabilities, inference_traces, uncertainty_annotations, provenance_records]
    safeguards: [preserve_conditioning_context, distinguish_probability_from_confidence, no_unsupported_causal_claims, validate_normalisation]
}

END FLOW ALGEBRA