Information Nessconditional Independence Measure Conditional Product Form

Conditional independence as a product

Conditional independence is a standard idea in probability: two events are conditionally independent given a third when, once you know the third, learning about one gives you no information about the other. The classical definition uses division: the probability of both events given the third must equal the product of their separate probabilities given the third. That definition fails when the conditioning event has probability zero, because dividing by zero is undefined.

The Recognition Science declaration conditional_product_form states the same idea in a form that avoids division entirely. It says that for any three coarse-grained coordinates, called internal, blanket, and external, the probability of all three together times the probability of the blanket alone equals the probability of internal and blanket together times the probability of blanket and external together. Written in symbols: P(I=i, B=b, E=e) · P(B=b) = P(I=i, B=b) · P(B=b, E=e). This product form is equivalent to the usual conditional independence statement whenever P(B=b) is not zero, and it remains a meaningful statement even when it is.

The declaration is a theorem in the framework's machine-checked library of formal theorems, not an assumption. It takes as input a probability measure and a projection of a state space into the three coordinates, plus a hypothesis that the measure satisfies the blanket factorization. From that hypothesis it derives the product identity for every choice of values. A companion theorem, ledger_sparsity_implies_measure_condIndep, shows that a property called ledger-boundary sparsity, which the framework treats as a measure of how cleanly a system separates into parts, directly implies the same conditional independence statement.

What the declaration does not claim is just as important. It does not say that any physical system actually satisfies the factorization; that is a separate hypothesis, not a consequence. It does not prove that the blanket coordinate is unique, or that the three-way split is the only possible one. It also does not connect this measure-theoretic statement to the framework's deeper results about recognition costs or the forcing chain; that connection remains a target for later work, explicitly noted in the source as a future derivation.

For a reader, the practical upshot is a clean, division-free way to state a common statistical assumption. The product form works even at events where the usual conditional probabilities would be undefined, which makes it a more solid foundation for building further theorems about how systems separate into internal states, surrounding blankets, and external environments.

/-- Product form of conditional independence. This is the event-level
conditional-independence identity, kept in multiplication form to avoid
division side conditions in `ENNReal`. -/
theorem conditional_product_form
    (P : ProbabilityMeasure Ω)
    (π : BlanketProjection Ω Internal Blanket External)
    (h : CondIndepGivenBlanket P π)
    (i : Internal) (b : Blanket) (e : External) :
    (P : Measure Ω) (atomSet π i b e) * (P : Measure Ω) (blanketSet π b) =
    (P : Measure Ω) (internalBlanketSet π i b) *
      (P : Measure Ω) (blanketExternalSet π b e) := by
  exact h i b e

/-- Product form of conditional independence. This is the event-level
conditional-independence identity, kept in multiplication form to avoid
division side conditions in `ENNReal`. -/
theorem conditional_product_form
    (P : ProbabilityMeasure Ω)
    (π : BlanketProjection Ω Internal Blanket External)
    (h : CondIndepGivenBlanket P π)
    (i : Internal) (b : Blanket) (e : External) :
    (P : Measure Ω) (atomSet π i b e) * (P : Measure Ω) (blanketSet π b) =
    (P : Measure Ω) (internalBlanketSet π i b) *
      (P : Measure Ω) (blanketExternalSet π b e) := by
  exact h i b e

theorem ledger_sparsity_implies_measure_condIndep
    (P : ProbabilityMeasure Ω)
    (π : BlanketProjection Ω Internal Blanket External)
    (h : LedgerBoundarySparsity P π) :
    CondIndepGivenBlanket P π := by
  exact h

What this page does not claim

The theorem does not assert that any physical system satisfies the factorization. The theorem does not prove uniqueness of the blanket coordinate or the three-way split. The theorem does not connect this statement to the framework's deeper results about recognition costs.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Information/NESSConditionalIndependenceMeasure.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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