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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 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 | import OAI.LinearAlgebra.MatrixMultiplication.Entropy.ComplexEntropyContinuity
namespace OAI
/-! Readable tensor completion and its finite arithmetic realization. -/
noncomputable section
namespace MatrixMultiplication.CompletionLaws
open MatrixMultiplication.Foundation
open scoped BigOperators
variable {A B X : Type*} [Fintype A] [Fintype B] [Fintype X]
def mixture (p : FiniteLaw B) (q : B → FiniteLaw A) : FiniteLaw A :=
(p.joint q).map Prod.snd
theorem mixture_mass (p : FiniteLaw B) (q : B → FiniteLaw A) (a : A) :
(mixture p q).mass a = ∑ b, p.mass b * (q b).mass a := by
classical
simp [mixture, FiniteLaw.map_mass, FiniteLaw.joint, Fintype.sum_prod_type]
theorem mixture_map_mass (p : FiniteLaw B) (q : B → FiniteLaw A)
(f : A → X) (x : X) :
((mixture p q).map f).mass x = ∑ b, p.mass b * ((q b).map f).mass x := by
classical
simp only [FiniteLaw.map_mass, mixture_mass]
calc
(∑ a, if f a = x then ∑ b, p.mass b * (q b).mass a else 0) =
∑ a, ∑ b, if f a = x then p.mass b * (q b).mass a else 0 := by
apply Finset.sum_congr rfl
intro a _
split_ifs <;> simp
_ = ∑ b, ∑ a, if f a = x then p.mass b * (q b).mass a else 0 :=
Finset.sum_comm
_ = ∑ b, p.mass b * ∑ a, if f a = x then (q b).mass a else 0 := by
apply Finset.sum_congr rfl
intro b _
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro a _
split_ifs <;> simp
theorem mixture_common_marginal (p : FiniteLaw B) (q : B → FiniteLaw A)
(f : A → X) (r : FiniteLaw X)
(hq : ∀ b x, ((q b).map f).mass x = r.mass x) (x : X) :
((mixture p q).map f).mass x = r.mass x := by
rw [mixture_map_mass]
simp_rw [hq]
rw [← Finset.sum_mul, p.total, one_mul]
theorem joint_common_marginal (p : FiniteLaw B) (q : B → FiniteLaw A)
(f : A → X) (r : FiniteLaw X)
(hq : ∀ b x, ((q b).map f).mass x = r.mass x) (b : B) (x : X) :
((p.joint q).map (fun ba => (ba.1, f ba.2))).mass (b, x) =
p.mass b * r.mass x := by
classical
simp only [FiniteLaw.map_mass, FiniteLaw.joint, Fintype.sum_prod_type,
Prod.mk.injEq]
calc
(∑ b', ∑ a, if b' = b ∧ f a = x then p.mass b' * (q b').mass a else 0) =
∑ a, if f a = x then p.mass b * (q b).mass a else 0 := by
rw [Finset.sum_eq_single b]
· simp
· intro b' _ hne
simp [hne]
· simp
_ = p.mass b * ((q b).map f).mass x := by
rw [FiniteLaw.map_mass, Finset.mul_sum]
apply Finset.sum_congr rfl
intro a _
split_ifs <;> simp
_ = _ := by rw [hq]
theorem common_marginal_entropy (p : FiniteLaw B) (q : B → FiniteLaw A)
(f : A → X) (r : FiniteLaw X)
(hq : ∀ b x, ((q b).map f).mass x = r.mass x) :
(∑ b, p.mass b * finiteEntropy ((q b).map f).mass) =
finiteEntropy r.mass := by
have heq (b : B) : ((q b).map f).mass = r.mass := funext (hq b)
simp_rw [heq]
rw [← Finset.sum_mul, p.total, one_mul]
theorem independent_entropy (p : FiniteLaw A) (q : FiniteLaw B) :
finiteEntropy (p.joint (fun _ => q)).mass =
finiteEntropy p.mass + finiteEntropy q.mass := by
rw [FiniteLaw.joint_entropy, ← Finset.sum_mul, p.total, one_mul]
end MatrixMultiplication.CompletionLaws
end
end OAI
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