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cdec422 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 | import OAI.LinearAlgebra.MatrixMultiplication.Completion.Conditioning
import OAI.LinearAlgebra.MatrixMultiplication.Duality.Information
namespace OAI
/-! Dual matrix multiplication exponents and finite rectangular constructions. -/
noncomputable section
namespace MatrixMultiplication.DualInformation
open MatrixMultiplication.Foundation CompletionLaws
attribute [local instance] Classical.propDecidable Classical.decEq
variable {A R L X : Type*} [Fintype A] [Fintype R] [Fintype L] [Fintype X]
theorem conditionalMassFactorization_condition (p : FiniteLaw A)
(prior : A → R) (label : A → L) (x : A → X) (Q : R → Prop)
(h : 0 < eventMass p (fun a => Q (prior a)))
(factor : ConditionalMassFactorization p prior label x) :
ConditionalMassFactorization (condition p (fun a => Q (prior a)) h)
(fun a => prior a.val) (fun a => label a.val) (fun a => x a.val) := by
intro r l xx
have hj := condition_coordinate_mass p (fun a => (prior a, (label a, x a)))
(fun v => Q v.1) h (r, (l, xx))
have hr := condition_coordinate_mass p prior Q h r
have hl := condition_coordinate_mass p (fun a => (prior a, label a))
(fun v => Q v.1) h (r, l)
have hx := condition_coordinate_mass p (fun a => (prior a, x a))
(fun v => Q v.1) h (r, xx)
change
((condition p (fun a => Q (prior a)) h).map
(fun a => (prior a.val, (label a.val, x a.val)))).mass (r, (l, xx)) *
((condition p (fun a => Q (prior a)) h).map (fun a => prior a.val)).mass r =
((condition p (fun a => Q (prior a)) h).map
(fun a => (prior a.val, label a.val))).mass (r, l) *
((condition p (fun a => Q (prior a)) h).map
(fun a => (prior a.val, x a.val))).mass (r, xx)
rw [hj, hr, hl, hx]
by_cases hQ : Q r
· simp only [hQ, ite_true]
rw [div_mul_div_comm, div_mul_div_comm, factor r l xx]
· simp only [hQ, ite_false, zero_mul]
theorem conditionalIndependent_condition_of_mass_factorization (p : FiniteLaw A)
(prior : A → R) (label : A → L) (x : A → X) (Q : R → Prop)
(h : 0 < eventMass p (fun a => Q (prior a)))
(factor : ConditionalMassFactorization p prior label x) :
ConditionalIndependent (condition p (fun a => Q (prior a)) h)
(fun a => prior a.val) (fun a => label a.val) (fun a => x a.val) :=
conditionalIndependent_of_mass_factorization _ _ _ _
(conditionalMassFactorization_condition p prior label x Q h factor)
theorem conditionalIndependent_condition (p : FiniteLaw A)
(prior : A → R) (label : A → L) (x : A → X) (Q : R → Prop)
(h : 0 < eventMass p (fun a => Q (prior a)))
(independent : ConditionalIndependent p prior label x) :
ConditionalIndependent (condition p (fun a => Q (prior a)) h)
(fun a => prior a.val) (fun a => label a.val) (fun a => x a.val) :=
conditionalIndependent_condition_of_mass_factorization p prior label x Q h
(mass_factorization_of_conditionalIndependent p prior label x independent)
end MatrixMultiplication.DualInformation
end
end OAI
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