Joint Probability Ontology

FLOW ALGEBRA
JOINT PROBABILITY ONTOLOGY
Version: 1.0
Date: 2026-10-13
Authors: Mark Higgins — Sky (Recursive AI)
Publisher: Kireth Research
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/probability-entities/joint-probability-ontology-2.html

PURPOSE
Formalise joint probability for events and random variables, preserving measurable-space assumptions, mathematical identities, semantic distinctions and machine-readable reasoning constraints.

ENTITY TYPES
ProbabilitySpace := (Omega, F, P)
Event := A in F
RandomVariable := measurable X: Omega -> S
JointEvent := Intersection(A,B)
JointDistribution := law(X,Y) on product measurable space
JointPMF := p_XY(x,y) for discrete variables
JointCDF := F_XY(x,y) = P(X <= x, Y <= y) for real-valued variables
JointDensity := f_XY(x,y) where a density exists
MarginalDistribution := law(X) or law(Y)
ConditionalDistribution := law(X | Y) where defined
DependenceRelation := Independent | Dependent | ConditionallyIndependent

FOUNDATIONAL AXIOMS
A1: P(Omega)=1.
A2: For any event A, P(A)>=0.
A3: For pairwise disjoint measurable events A_i, P(union_i A_i)=sum_i P(A_i).
A4: Joint event probability P(A,B) denotes P(A intersection B).
A5: 0 <= P(A intersection B) <= min(P(A),P(B)).
A6: P(A intersection B)=P(B intersection A).

OPERATORS AND IDENTITIES
JOINT_EVENT(A,B) := A intersection B.
JOINT_PROBABILITY(A,B) := P(A intersection B).
PRODUCT_RULE(A,B) := P(A|B)*P(B), provided P(B)>0.
REVERSE_PRODUCT_RULE(A,B) := P(B|A)*P(A), provided P(A)>0.
INDEPENDENCE(A,B) := P(A intersection B)=P(A)*P(B).
UNION_RULE(A,B) := P(A union B)=P(A)+P(B)-P(A intersection B).
JOINT_COMPLEMENT(A,B) := P(A intersection complement(B))=P(A)-P(A intersection B).

DISCRETE RANDOM VARIABLES
p_XY(x,y) := P(X=x and Y=y).
p_X(x) := sum_y p_XY(x,y).
p_Y(y) := sum_x p_XY(x,y).
sum_x sum_y p_XY(x,y)=1.
p_XY(x,y)=p_X(x)*p_Y(y) for independent X,Y.
For p_Y(y)>0: p_X_given_Y(x|y)=p_XY(x,y)/p_Y(y).

CONTINUOUS RANDOM VARIABLES
If a joint density exists: f_XY(x,y)>=0 almost everywhere.
Integral over R^2 of f_XY(x,y) dx dy = 1.
f_X(x) := integral over R f_XY(x,y) dy.
f_Y(y) := integral over R f_XY(x,y) dx.
P((X,Y) in B) := integral over B f_XY(x,y) dx dy for measurable B.
F_XY(x,y) := P(X<=x,Y<=y).
Where densities exist and f_Y(y)>0: f_X_given_Y(x|y)=f_XY(x,y)/f_Y(y).
A density value is not itself a point probability.

GENERAL JOINT DISTRIBUTION
For measurable sets C subset S and D subset T:
P_XY(C x D) := P(X in C and Y in D).
Marginals are pushforwards of the joint law under coordinate projections.
Not every joint distribution has a joint density or joint PMF.
Conditional distributions may require regular conditional probability assumptions.

TYPED RELATIONSHIPS
RandomExperiment --defines--> ProbabilitySpace
ProbabilitySpace --contains--> Event
Event --intersects_with--> Event
JointEvent --has_probability--> ProbabilityValue
RandomVariable --is_measurable_on--> ProbabilitySpace
JointDistribution --couples--> RandomVariable
JointDistribution --has_marginal--> MarginalDistribution
JointDistribution --may_have_representation--> JointPMF
JointDistribution --may_have_representation--> JointDensity
JointDistribution --supports--> ConditionalDistribution
IndependenceRelation --factorises--> JointDistribution

INFERENCE RULES
R1: IF A,B are measurable THEN A intersection B is measurable.
R2: IF P(B)>0 THEN P(A intersection B)=P(A|B)P(B).
R3: IF A and B are independent THEN P(A intersection B)=P(A)P(B).
R4: IF a valid discrete joint PMF is supplied THEN marginalise by summation.
R5: IF a valid joint density is supplied THEN marginalise by integration.
R6: IF joint law factorises into its marginals on all measurable rectangles THEN X and Y are independent.
R7: IF joint probability is zero THEN do not infer independence without additional evidence.
R8: IF P(A intersection B)>0 THEN neither A nor B has probability zero.

SEMANTIC DISTINCTIONS
Joint probability of events != conditional probability.
Joint distribution != marginal distribution.
Independence != mutual exclusivity (except special zero-probability cases).
Joint density != probability mass.
Correlation zero != independence in general.
Joint distribution != causal relationship.

VALIDATION
V1: Check probability-space and measurability prerequisites.
V2: Check event probability lies in [0,1].
V3: Check PMF entries are nonnegative and sum to one.
V4: Check density is nonnegative almost everywhere and integrates to one.
V5: Check marginalisation recovers valid marginal laws.
V6: Check conditional denominators are positive before division.
V7: Check independence claims against factorisation, not correlation alone.
V8: Do not assume a density exists for arbitrary random variables.

CONFIDENCE
Formal mathematical identity with satisfied assumptions: 1.0.
Empirical joint probability estimate: attach estimator, sample size, uncertainty and provenance; do not assign certainty automatically.

SEMANTIC_EQUIVALENCE
"joint event probability" ~ "probability of intersection" when events are specified.
"bivariate distribution" ~ "joint distribution of two random variables".
"joint PMF" != "joint PDF".

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AI_CONSUMPTION
Parse typed entities before applying identities.
Verify domains, measurable sets, existence of PMFs/densities and denominator conditions.
Separate exact symbolic results from empirical estimates.
Return derivations with assumptions, confidence, provenance and any unresolved constraints.
Do not infer causality from a joint distribution.

END FLOW ALGEBRA