| import OAI.LinearAlgebra.MatrixMultiplication.Entropy.ComplexEntropyContinuity |
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| namespace OAI |
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| /-! Readable tensor completion and its finite arithmetic realization. -/ |
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| noncomputable section |
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| namespace MatrixMultiplication.CompletionLaws |
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| open MatrixMultiplication.Foundation |
| open scoped BigOperators |
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| variable {A B X : Type*} [Fintype A] [Fintype B] [Fintype X] |
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| def mixture (p : FiniteLaw B) (q : B → FiniteLaw A) : FiniteLaw A := |
| (p.joint q).map Prod.snd |
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| 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] |
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| 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] |
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| 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] |
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| 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] |
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| 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] |
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| end MatrixMultiplication.CompletionLaws |
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| end |
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| end OAI |
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