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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.Completion.Laws
import OAI.LinearAlgebra.MatrixMultiplication.Entropy.ComplexHierarchyEntropy
namespace OAI
/-! Dual matrix multiplication exponents and finite rectangular constructions. -/
noncomputable section
namespace MatrixMultiplication.DualEntropyHelpers
open MatrixMultiplication.Foundation
open scoped BigOperators
variable {A B C X : Type*} [Fintype A] [Fintype B] [Fintype C] [Fintype X]
theorem map_comp_mass (p : FiniteLaw A) (f : A → B) (g : B → C) :
((p.map f).map g).mass = (p.map (fun a => g (f a))).mass := by
classical
funext c
simp only [FiniteLaw.map_mass]
calc
(∑ b, if g b = c then ∑ a, if f a = b then p.mass a else 0 else 0) =
∑ b, ∑ a, if f a = b then (if g b = c then p.mass a else 0) else 0 := by
apply Finset.sum_congr rfl
intro b _
by_cases h : g b = c <;> simp [h]
_ = ∑ a, ∑ b, if f a = b then (if g b = c then p.mass a else 0) else 0 :=
Finset.sum_comm
_ = ∑ a, if g (f a) = c then p.mass a else 0 := by
apply Finset.sum_congr rfl
intro a _
simp
theorem deterministic_pair_entropy (p : FiniteLaw A) (f : A → B) (g : B → C) :
finiteEntropy (p.map (fun a => (f a, g (f a)))).mass =
finiteEntropy (p.map f).mass := by
rw [← map_comp_mass p f (fun b => (b, g b))]
exact (p.map f).map_entropy_of_injective _ (fun _ _ h => congrArg Prod.fst h)
theorem pair_entropy (p : FiniteLaw A) (f : A → B) (g : A → C) :
finiteEntropy (p.map (fun a => (f a, g a))).mass =
finiteEntropy (p.map f).mass +
∑ b, (p.map f).mass b * finiteEntropy ((p.conditional f b).map g).mass := by
have hm : ((p.map f).joint (fun b => (p.conditional f b).map g)).mass =
(p.map (fun a => (f a, g a))).mass := by
funext bc
exact p.map_mass_mul_conditional_map f g bc.1 bc.2
rw [← hm, FiniteLaw.joint_entropy]
theorem conditional_independence_entropy (p : FiniteLaw A)
(f : A → B) (g : A → C) (x : A → X)
(independent : ∀ b, 0 < (p.map f).mass b → ∀ c xx,
((p.conditional f b).map (fun a => (g a, x a))).mass (c, xx) =
((p.conditional f b).map g).mass c *
((p.conditional f b).map x).mass xx) :
finiteEntropy (p.map (fun a => ((f a, g a), x a))).mass -
finiteEntropy (p.map (fun a => (f a, g a))).mass =
finiteEntropy (p.map (fun a => (f a, x a))).mass -
finiteEntropy (p.map f).mass := by
classical
have hassoc : finiteEntropy (p.map (fun a => ((f a, g a), x a))).mass =
finiteEntropy (p.map (fun a => (f a, (g a, x a)))).mass := by
have h := (p.map (fun a => ((f a, g a), x a))).map_entropy_of_injective
(Equiv.prodAssoc B C X) (Equiv.prodAssoc B C X).injective
rw [map_comp_mass] at h
exact h.symm
have hcond (b : B) :
(p.map f).mass b *
finiteEntropy ((p.conditional f b).map (fun a => (g a, x a))).mass =
(p.map f).mass b *
(finiteEntropy ((p.conditional f b).map g).mass +
finiteEntropy ((p.conditional f b).map x).mass) := by
by_cases hb : (p.map f).mass b = 0
· simp only [hb, zero_mul]
· have hpos : 0 < (p.map f).mass b :=
lt_of_le_of_ne ((p.map f).nonneg b) (Ne.symm hb)
have hm : ((p.conditional f b).map (fun a => (g a, x a))).mass =
(((p.conditional f b).map g).joint
(fun _ => (p.conditional f b).map x)).mass := by
funext cx
exact independent b hpos cx.1 cx.2
rw [hm, CompletionLaws.independent_entropy]
rw [hassoc, pair_entropy p f (fun a => (g a, x a)),
pair_entropy p f g, pair_entropy p f x]
simp_rw [hcond]
simp only [mul_add, Finset.sum_add_distrib]
ring
end MatrixMultiplication.DualEntropyHelpers
end
end OAI
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