Probability Entity Ontology

FLOW ALGEBRA
PROBABILITY ENTITY ONTOLOGY
Kireth Semantic Library | Entity Graph > Mathematical Entities > Probability Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/probability-entities/probability-entity-ontology.html
Version: 1.0 | 2026-10-13 | Locale: en-GB
Authors: Mark Higgins — Sky (Recursive AI)

1. PURPOSE
Represent probability-related mathematical entities as typed, verifiable semantic objects with explicit assumptions, relationships, reasoning rules and provenance.

2. CORE TYPES
entity ProbabilityEntity : MathematicalEntity
entity ProbabilitySpace : ProbabilityEntity {sample_space: Set, sigma_algebra: SigmaAlgebra, measure: ProbabilityMeasure}
entity Outcome : ProbabilityEntity {member_of: SampleSpace}
entity Event : ProbabilityEntity {subset_of: SampleSpace, measurable: Boolean}
entity ProbabilityMeasure : ProbabilityEntity {domain: SigmaAlgebra, codomain: RealInterval[0,1]}
entity RandomVariable : ProbabilityEntity {domain: SampleSpace, codomain: MeasurableSpace, measurable: Boolean}
entity Distribution : ProbabilityEntity {associated_with: RandomVariable}
entity ConditionalProbability : ProbabilityEntity {event: Event, given: Event, denominator_positive: Boolean}
entity Expectation : ProbabilityEntity {random_variable: RandomVariable, integrable: Boolean}
entity Variance : ProbabilityEntity {random_variable: RandomVariable, finite_second_moment: Boolean}
entity Independence : ProbabilityEntity {events: Set[Event]}

3. FORMAL FOUNDATIONS
Let (Omega, F, P) be a probability space.
F is a sigma-algebra on Omega.
P: F -> [0,1].
Axiom P1: P(A) >= 0 for every A in F.
Axiom P2: P(Omega) = 1.
Axiom P3: For pairwise disjoint A_1,A_2,... in F,
          P(union_n A_n) = sum_n P(A_n).
Derived: P(empty_set) = 0.
Derived: P(A_complement) = 1 - P(A).
Derived: P(A union B) = P(A) + P(B) - P(A intersection B).
Derived: A subseteq B => P(A) <= P(B).

4. OPERATORS
probability(event: Event, space: ProbabilitySpace) -> RealInterval[0,1]
complement(event: Event) -> Event
intersection(event_a: Event, event_b: Event) -> Event
union(event_a: Event, event_b: Event) -> Event
condition(event_a: Event, given_b: Event) -> ConditionalProbability
expectation(variable: RandomVariable) -> ExtendedReal
variance(variable: RandomVariable) -> NonnegativeReal
pushforward(variable: RandomVariable, measure: ProbabilityMeasure) -> Distribution

5. CONDITIONAL PROBABILITY AND BAYES
If P(B) > 0: P(A | B) = P(A intersection B) / P(B).
If P(A) > 0 and P(B) > 0:
P(A | B) = P(B | A) P(A) / P(B).
Events A and B are independent iff P(A intersection B) = P(A) P(B).
Pairwise independence does not imply mutual independence.
Conditioning on an event of probability zero requires additional formal machinery (e.g. a regular conditional distribution), not the elementary quotient.

6. RANDOM VARIABLES AND MOMENTS
A random variable X: Omega -> S must be measurable.
Distribution of X: P_X(B) = P(X inverse_image(B)).
For real-valued integrable X: E[X] = integral_Omega X dP.
For E[X^2] finite: Var(X) = E[(X-E[X])^2] = E[X^2] - E[X]^2.
Linearity: E[aX+bY] = aE[X]+bE[Y] when expectations exist.
If X,Y are independent and square-integrable, Cov(X,Y)=0; converse is generally false.

7. TYPED RELATIONSHIPS
ProbabilitySpace --HAS_SAMPLE_SPACE--> SampleSpace
ProbabilitySpace --HAS_SIGMA_ALGEBRA--> SigmaAlgebra
ProbabilitySpace --HAS_MEASURE--> ProbabilityMeasure
Event --IS_MEASURABLE_SUBSET_OF--> SampleSpace
Outcome --BELONGS_TO--> SampleSpace
ProbabilityMeasure --ASSIGNS_PROBABILITY_TO--> Event
RandomVariable --MAPS_FROM--> SampleSpace
RandomVariable --INDUCES--> Distribution
ConditionalProbability --CONDITIONS_ON--> Event
Expectation --SUMMARISES--> RandomVariable
Variance --MEASURES_DISPERSION_OF--> RandomVariable
Independence --RELATES--> Event

8. INFERENCE RULES
R1: measurable(A) AND measurable(B) => measurable(A union B).
R2: measurable(A) => measurable(complement(A)).
R3: disjoint(A,B) => P(A union B) = P(A)+P(B).
R4: subset(A,B) => P(A) <= P(B).
R5: P(B)>0 => P(A|B)=P(A intersection B)/P(B).
R6: independent(A,B) => P(A|B)=P(A), provided P(B)>0.
R7: finite_second_moment(X) => Var(X)>=0.
R8: independent(X,Y) AND finite_second_moments(X,Y) => Cov(X,Y)=0.
Each rule must record its premises and refuse conclusions when preconditions fail.

9. SEMANTIC DISTINCTIONS
Outcome != Event: an outcome is a sample-space element; an event is a measurable subset.
Probability measure != probability density: a density is a representation relative to a reference measure, when it exists.
Distribution != random variable: distinct variables can share a distribution.
Independence != mutual exclusivity: disjoint positive-probability events are not independent.
Probability != confidence score: a model score is probabilistic only if its interpretation and calibration are established.
Probability entity != observed frequency: empirical estimates require sample design and uncertainty metadata.

10. VALIDATION
V1: Probability values lie in [0,1].
V2: Event belongs to the declared sigma-algebra.
V3: Normalisation P(Omega)=1.
V4: Countable additivity is preserved.
V5: Conditional probabilities require a positive-probability conditioning event for quotient semantics.
V6: Random variables satisfy measurability requirements.
V7: Moments and variances declare existence/finite-moment assumptions.
V8: Inference records distinguish theorem, assumption, estimate and conjecture.
V9: Never infer independence from zero covariance without extra assumptions.

11. CONFIDENCE AND PROVENANCE
CONFIDENCE: {value: [0,1], interpretation: evidence_quality, basis: source_and_validation}
PROVENANCE: {source_url, author, version, timestamp, assumptions, derivation, validation_status}
SEMANTIC_EQUIVALENCE: only after matching domains, measurable structures, assumptions and meaning.

12. FLOW HASH AND UPDATE MODEL
FLOW_HASH: SHA-256 of canonical UTF-8 representation of ontology identifier, version, typed entities, relationships, rules and assumptions; computed by the implementing system, not preclaimed here.
UPDATE_MODEL: versioned changes; preserve prior versions; recalculate hash after material edits; revalidate dependent entities and inference rules.

13. AI CONSUMPTION
AI_CONSUMPTION:
  ingest: typed_entities + axioms + operators + relationships + rules
  resolve: entity_identity + mathematical_context + assumptions
  reason: precondition_checked_inference
  output: claims + supporting_rules + provenance + confidence
  reject: ill_typed_operations + unsupported_independence + invalid_conditioning
  audit: version + canonical_hash + validation_results

14. EXAMPLE
Let Omega={H,T}, F=power_set(Omega), P({H})=P({T})=0.5.
A={H}; P(A)=0.5; P(A_complement)=0.5.
Check: P(Omega)=1 and P(empty_set)=0.
Classification: A is an Event; H is an Outcome; P is a ProbabilityMeasure.

END FLOW ALGEBRA