Boolean Markovian Response Law Characterization

Abstract

A normalized nonnegative law on Bool x Bool is a product of two coordinate laws exactly when its two-by-two determinant vanishes.

Theorem 1.1 (Product structure is exactly determinant vanishing).

$$\forall P,\mathrm{isMarkovianTwoComponentLaw}(P)⟺P(tt)\cdot P(ff)=P(tf)\cdot P(ft).$$

Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/PartialIdentification/BooleanMarkovianResponseLawCharacterization.boolean_markovian_iff_determinant_zero (✓ std3). ∎

Source. Repository-derived.

Commentary.

Necessity is the product determinant identity. For sufficiency, the two coordinate marginals are taken; normalization and the determinant equation show cell by cell that their product reconstructs the law.

This is the two-mode boundary case of the partial identification programme: independence of a joint Boolean response law is a single polynomial constraint on its four masses.

References

• Truth anchor: D5/S3/ConceptDynamics/PartialIdentification/BooleanMarkovianResponseLawCharacterization.boolean_markovian_iff_determinant_zero
• Dependency: D5/S3/ConceptDynamics/PartialIdentification/MarkovianBenefitIdentificationBoundary