CondIndepGivenBlanket

formal source

  54  {ω | π.blanket ω = b}
  55
  56/-- Measure-theoretic conditional independence as blanket factorization. -/
  57def CondIndepGivenBlanket
  58    (P : ProbabilityMeasure Ω)
  59    (π : BlanketProjection Ω Internal Blanket External) : Prop :=
  60  ∀ (i : Internal) (b : Blanket) (e : External),
  61    (P : Measure Ω) (atomSet π i b e) * (P : Measure Ω) (blanketSet π b) =
  62    (P : Measure Ω) (internalBlanketSet π i b) *
  63      (P : Measure Ω) (blanketExternalSet π b e)
  64
  65/-- Ledger-boundary sparsity on the measure surface: the measure has the
  66blanket factorization. In later work this can be derived from a concrete
  67recognition-field generator; here it is the exact hypothesis needed for
  68conditional independence. -/
  69def LedgerBoundarySparsity
  70    (P : ProbabilityMeasure Ω)
  71    (π : BlanketProjection Ω Internal Blanket External) : Prop :=
  72  ∀ (i : Internal) (b : Blanket) (e : External),
  73    (P : Measure Ω) (atomSet π i b e) * (P : Measure Ω) (blanketSet π b) =
  74    (P : Measure Ω) (internalBlanketSet π i b) *
  75      (P : Measure Ω) (blanketExternalSet π b e)
  76
  77theorem ledger_sparsity_implies_measure_condIndep
  78    (P : ProbabilityMeasure Ω)
  79    (π : BlanketProjection Ω Internal Blanket External)
  80    (h : LedgerBoundarySparsity P π) :
  81    CondIndepGivenBlanket P π := by
  82  exact h
  83
  84/-- Product form of conditional independence. This is the event-level
  85conditional-independence identity, kept in multiplication form to avoid
  86division side conditions in `ENNReal`. -/
  87theorem conditional_product_form