Variance Ontology

FLOW ALGEBRA
VARIANCE ONTOLOGY
Kireth Semantic Library | Mathematical Entities > Probability Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/probability-entities/variance-ontology-2.html
Version: 1.0 | Date: 2026-10-13
Authors: Mark Higgins — Sky (Recursive AI)

1. PURPOSE
Represent variance as a typed mathematical entity describing dispersion of a real-valued random variable about its expectation, with computable definitions, guarded inference rules, and links to probability theory and statistics.

2. ENTITY TYPES
ProbabilitySpace := (Omega, SigmaAlgebra, ProbabilityMeasure)
RandomVariable := measurable mapping X: Omega -> R
Expectation := E[X], provided integrability
Variance := Var(X) in [0,+infinity), provided E[X^2] < infinity
StandardDeviation := sqrt(Var(X))
Covariance := Cov(X,Y), provided finite second moments
ConditionalVariance := Var(X | G), conditional on sub-sigma-algebra G
SampleVariance := statistic derived from observations, distinct from population variance

3. DEFINITIONS AND AXIOMS
For X with E[X^2] < infinity, let mu = E[X].
VAR(X) = E[(X-mu)^2] = E[X^2] - (E[X])^2.
VAR(X) >= 0.
VAR(X) = 0 iff X is constant almost surely.
VAR(aX+b) = a^2 VAR(X) for real constants a,b.
SD(X) = sqrt(VAR(X)).
For X,Y with finite second moments:
VAR(X+Y) = VAR(X)+VAR(Y)+2 COV(X,Y).
VAR(X-Y) = VAR(X)+VAR(Y)-2 COV(X,Y).
If X,Y are independent, COV(X,Y)=0 and VAR(X+Y)=VAR(X)+VAR(Y).
The converse need not hold.
For independent Xi with finite variances, VAR(sum_i Xi)=sum_i VAR(Xi).

4. DISCRETE AND CONTINUOUS FORMS
Discrete X with PMF p(x): mu=sum_x x*p(x); VAR(X)=sum_x (x-mu)^2*p(x).
Absolutely continuous X with PDF f(x): mu=integral_R x*f(x) dx; VAR(X)=integral_R (x-mu)^2*f(x) dx.
These expressions require finite second moments. Not every continuous distribution has a PDF.

5. SAMPLE STATISTICS
For observations x1,...,xn, n>=2:
xbar=(1/n)*sum_i xi.
s2=(1/(n-1))*sum_i (xi-xbar)^2, the usual unbiased sample variance under IID sampling with finite population variance.
Population-style empirical variance uses divisor n and is not the same estimator.

6. CONDITIONAL AND TOTAL VARIANCE
VAR(X | G)=E[(X-E[X|G])^2 | G].
Law of total variance: VAR(X)=E[VAR(X|G)]+VAR(E[X|G]).
Requires finite second moment; conditional variance is G-measurable.

7. TYPED RELATIONSHIPS
Variance --measures_dispersion_of--> RandomVariable
Variance --computed_from--> Expectation
Variance --requires--> FiniteSecondMoment
Variance --square_root_is--> StandardDeviation
Variance --generalises_with--> Covariance
Variance --conditional_form--> ConditionalVariance
SampleVariance --estimates--> PopulationVariance
Variance --belongs_to--> ProbabilityAndStatistics

8. INFERENCE RULES
RULE V01: IF finite_second_moment(X) THEN defined(VAR(X)).
RULE V02: IF defined(VAR(X)) THEN VAR(X)>=0.
RULE V03: IF defined(VAR(X)) AND a,b real THEN VAR(aX+b)=a*a*VAR(X).
RULE V04: IF finite_second_moments(X,Y) AND independent(X,Y) THEN VAR(X+Y)=VAR(X)+VAR(Y).
RULE V05: IF VAR(X)=0 AND defined(VAR(X)) THEN constant_almost_surely(X).
RULE V06: IF finite_second_moment(X) THEN VAR(X)=E[X^2]-E[X]^2.
RULE V07: IF finite_second_moment(X) AND sub_sigma_algebra(G) THEN total_variance_identity(X,G).

9. SEMANTIC DISTINCTIONS
Variance is not standard deviation: their units are squared versus original units.
Variance is not covariance: covariance concerns two random variables.
Zero covariance does not generally imply independence.
Population variance is not automatically interchangeable with sample variance.
Infinite or undefined variance must not be silently represented as a finite value.

10. VALIDATION
CHECK probability_measure_is_normalised.
CHECK random_variable_is_measurable.
CHECK second_moment_is_finite_before_finite_variance_calculation.
CHECK computed_variance_is_nonnegative_with_numerical_tolerance.
CHECK sample_variance_requires_n_at_least_2.
CHECK independence_is_not_inferred_from_zero_covariance.
CHECK units_of_variance_are_squared_units_of_X.

11. CONFIDENCE AND PROVENANCE
CONFIDENCE: 1.0 for definitions/theorems under explicitly satisfied assumptions; otherwise report assumption status.
PROVENANCE: Record source ontology URL, mathematical premises, computation method, and revision date.
SEMANTIC_EQUIVALENCE: Variance = second central moment when finite; not equivalent to standard deviation.

12. FLOW HASH AND UPDATE MODEL
FLOW_HASH: Compute SHA-256 over the canonical UTF-8 serialisation of this flow, excluding the FLOW_HASH value itself; no digest is asserted until computed.
UPDATE_MODEL: versioned; preserve identifiers, track axiom changes, recalculate hash after updates, and retain provenance.

13. AI CONSUMPTION
INPUT: typed random variable or numerical observations; probability model; assumptions.
PROCESS: validate model -> test moments -> select formula -> compute/derive -> check units and domain -> attach provenance.
OUTPUT: variance value or symbolic expression, definedness status, assumptions, confidence, and linked entities.
FAILURE: report undefined/infinite variance, missing distribution, insufficient sample size, or violated assumptions without fabricating a numeric result.

END FLOW ALGEBRA