atomSet

formal source

  34  external : Ω → External
  35
  36/-- Event where all three coarse-grained coordinates take specified values. -/
  37def atomSet (π : BlanketProjection Ω Internal Blanket External)
  38    (i : Internal) (b : Blanket) (e : External) : Set Ω :=
  39  {ω | π.internal ω = i ∧ π.blanket ω = b ∧ π.external ω = e}
  40
  41/-- Event where internal and blanket coordinates take specified values. -/
  42def internalBlanketSet (π : BlanketProjection Ω Internal Blanket External)
  43    (i : Internal) (b : Blanket) : Set Ω :=
  44  {ω | π.internal ω = i ∧ π.blanket ω = b}
  45
  46/-- Event where blanket and external coordinates take specified values. -/
  47def blanketExternalSet (π : BlanketProjection Ω Internal Blanket External)
  48    (b : Blanket) (e : External) : Set Ω :=
  49  {ω | π.blanket ω = b ∧ π.external ω = e}
  50
  51/-- Blanket event. -/
  52def blanketSet (π : BlanketProjection Ω Internal Blanket External)
  53    (b : Blanket) : Set Ω :=
  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