Conditional Probability Ontology

FLOW ALGEBRA
TITLE: Conditional Probability Ontology
VERSION: 4.0
DOMAIN: Mathematical Entities / Probability Entities
SOURCE: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/probability-entities/conditional-probability-ontology-2.html
AUTHORS: Mark Higgins — Sky (Recursive AI)
DATE: 2026-10-13

PURPOSE
Represent conditional probability as a typed, verifiable mathematical entity with explicit prerequisites, identities, inference rules, and semantic links for AI reasoning.

ENTITY TYPES
ProbabilitySpace := (Omega, SigmaAlgebra, ProbabilityMeasure)
Event := measurable subset of Omega
ConditioningEvent := Event B with P(B) > 0
ConditionalProbability := P(A | B), for events A,B and P(B)>0
ConditionalDistribution := distribution under conditioning information
Partition := countable disjoint measurable events covering Omega
IndependenceRelation := relation between events satisfying P(A intersect B)=P(A)P(B)

CORE DEFINITIONS
P(A | B) = P(A intersect B) / P(B), provided P(B)>0.
P(A intersect B) = P(A | B) P(B), provided P(B)>0.
For fixed B with P(B)>0, P_B(A) := P(A | B) is a probability measure on the original sigma-algebra.
P(Omega | B)=1; P(empty | B)=0; 0<=P(A | B)<=1.
P(A complement | B)=1-P(A | B).
If A subset C then P(A | B)<=P(C | B).
For disjoint A_i: P(union_i A_i | B)=sum_i P(A_i | B).

BAYES AND TOTAL PROBABILITY
Bayes: P(A | B)=P(B | A)P(A)/P(B), when P(A)>0 and P(B)>0.
For a finite/countable measurable partition (H_i) with positive probabilities:
P(B)=sum_i P(B | H_i)P(H_i).
P(H_j | B)=P(B | H_j)P(H_j)/sum_i P(B | H_i)P(H_i), when P(B)>0.
Zero-probability partition cells must be omitted or treated using an appropriately defined conditional law.

INDEPENDENCE
A and B are independent iff P(A intersect B)=P(A)P(B).
When P(B)>0, independence implies P(A | B)=P(A).
Pairwise independence does not imply mutual independence for three or more events.
Conditional independence of A,C given B (P(B)>0):
P(A intersect C | B)=P(A | B)P(C | B).
Conditional independence need not imply unconditional independence, nor vice versa.

CHAIN RULE
For events A1,...,An with all required prefix-intersection probabilities positive:
P(intersection_i Ai)=P(A1) product_{k=2..n} P(Ak | intersection_{j<k} Aj).

TYPED RELATIONSHIPS
ConditionalProbability --conditions_on--> ConditioningEvent
ConditionalProbability --evaluates--> Event
ConditionalProbability --derived_from--> ProbabilityMeasure
ConditioningEvent --belongs_to--> SigmaAlgebra
Event --subset_of--> SampleSpace
BayesRule --reverses_conditioning_between--> Events
TotalProbability --marginalises_over--> Partition
IndependenceRelation --constrains--> JointProbability
ConditionalDistribution --generalises--> ConditionalProbability

INFERENCE RULES
R1: IF measurable(A), measurable(B), P(B)>0 THEN define P(A|B)=P(A intersect B)/P(B).
R2: IF P(B)>0 AND P(A intersect B)=P(A)P(B) THEN infer P(A|B)=P(A).
R3: IF P(A)>0 AND P(B)>0 THEN infer Bayes identity.
R4: IF (H_i) is a measurable partition AND P(B)>0 THEN infer posterior probabilities from total probability, subject to defined terms.
R5: IF A subset C AND P(B)>0 THEN infer P(A|B)<=P(C|B).
R6: IF P(B)=0 THEN flag elementary event-ratio conditional probability as undefined; do not divide by zero.
R7: IF A_i are pairwise disjoint AND P(B)>0 THEN infer countable conditional additivity.

SEMANTIC DISTINCTIONS
Conditional probability is not the same as joint probability.
P(A|B) generally differs from P(B|A).
Conditioning on a zero-probability event requires additional mathematical structure (e.g. regular conditional probabilities), not the elementary ratio formula.
Correlation, causation, and conditional dependence are distinct concepts.
Conditional probability given an event differs from conditioning on a sigma-algebra or random variable, although the concepts are related.

VALIDATION
V1: Verify probability-space axioms and event measurability.
V2: Check P(B)>0 before applying the event-ratio definition.
V3: Check all results are within [0,1].
V4: Verify normalization and countable additivity for fixed valid B.
V5: Check Bayes and partition preconditions before inference.
V6: Preserve conditioning direction and provenance.
V7: Do not infer causality from conditioning alone.

CONFIDENCE
Mathematical identities: deductive, confidence 1.0 when assumptions are verified.
Empirical probability estimates: attach estimation method, sample size, uncertainty and provenance; never automatically assign confidence 1.0.

SEMANTIC_EQUIVALENCE
"probability of A given B" == "P(A|B)" under the same probability space and conditioning event.
"posterior probability" may denote a conditional probability in Bayesian inference but is not universally synonymous with every conditional probability.

FLOW_HASH
canonical_payload := normalize(VERSION, TITLE, ENTITY_TYPES, CORE_DEFINITIONS, RELATIONSHIPS, INFERENCE_RULES, VALIDATION)
FLOW_HASH := SHA256(UTF8(canonical_payload))
Implementation note: calculate hash on the final canonical serialization; no hash value is asserted here.

UPDATE_MODEL
On change: increment version; preserve source and previous hash; rerun validation; record changed assumptions, definitions and relationships.

AI_CONSUMPTION
Input: typed probability space, measurable events, conditioning context, known probabilities.
Process: validate types and assumptions -> select applicable identity -> compute/infer -> verify range and normalization -> record provenance.
Output: conditional probability or symbolic expression, conditions, inference trace, uncertainty where estimated, and validation status.
On invalid inputs: return an explicit undefined/insufficient-information status rather than fabricate a probability.

END FLOW ALGEBRA