| import OAI.Computability.UniqueGames.Inverse.EqualityFiber |
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| namespace OAI |
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| section |
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| namespace UniqueGamesTheorem.Inverse.FactorSampling |
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| noncomputable section |
| open scoped BigOperators Classical |
| open UniqueGamesTheorem.Foundations.Information |
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| variable {A B : Type*} [Fintype A] [Fintype B] [Zero A] [Zero B] |
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| def fullAverage (p : A → ℝ) (q : B → ℝ) (h : A → B → ℝ) : ℝ := |
| ∑ a, ∑ b, p a * q b * h a b |
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| def nonzeroFlow (p : A → ℝ) (q : B → ℝ) (h : A → B → ℝ) : ℝ := |
| ∑ a, ∑ b, p a * q b * if a ≠ 0 ∧ b ≠ 0 then h a b else 0 |
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| def zeroMass (p : A → ℝ) (q : B → ℝ) : ℝ := |
| p 0 + q 0 - p 0 * q 0 |
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| omit [Fintype A] [Fintype B] in |
| private theorem boundary_indicator (a : A) (b : B) : |
| (if a = 0 ∨ b = 0 then (1 : ℝ) else 0) = |
| (if a = 0 then 1 else 0) + (if b = 0 then 1 else 0) - |
| (if a = 0 then 1 else 0) * (if b = 0 then 1 else 0) := by |
| by_cases ha : a = 0 <;> by_cases hb : b = 0 <;> simp [ha, hb] |
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| / |
| theorem boundary_mass (p : A → ℝ) (q : B → ℝ) |
| (hp : IsProbability p) (hq : IsProbability q) : |
| fullAverage p q (fun a b => if a = 0 ∨ b = 0 then 1 else 0) = zeroMass p q := by |
| unfold fullAverage zeroMass |
| simp_rw [boundary_indicator, mul_sub, mul_add] |
| simp only [Finset.sum_sub_distrib, Finset.sum_add_distrib] |
| have h₁ : (∑ a, ∑ b, p a * q b * if a = 0 then 1 else 0) = p 0 := by |
| calc |
| _ = ∑ a, p a * (∑ b, q b) * if a = 0 then 1 else 0 := by |
| apply Finset.sum_congr rfl |
| intro a _ |
| rw [← Finset.sum_mul, ← Finset.mul_sum] |
| _ = p 0 := by simp [hq.2, mul_ite] |
| have h₂ : (∑ a, ∑ b, p a * q b * if b = 0 then 1 else 0) = q 0 := by |
| simp only [mul_ite, mul_one, mul_zero, Finset.sum_ite_eq', Finset.mem_univ, ite_true] |
| rw [← Finset.sum_mul, hp.2, one_mul] |
| have h₃ : (∑ a, ∑ b, p a * q b * |
| ((if a = 0 then 1 else 0) * (if b = 0 then 1 else 0))) = p 0 * q 0 := by |
| simp [mul_ite] |
| rw [h₁, h₂, h₃] |
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| / |
| theorem fullAverage_eq_zeroMass_add_nonzeroFlow (p : A → ℝ) (q : B → ℝ) |
| (hp : IsProbability p) (hq : IsProbability q) (h : A → B → ℝ) |
| (hleft : ∀ b, h 0 b = 1) (hright : ∀ a, h a 0 = 1) : |
| fullAverage p q h = zeroMass p q + nonzeroFlow p q h := by |
| have hpoint (a : A) (b : B) : h a b = |
| (if a = 0 ∨ b = 0 then 1 else 0) + |
| (if a ≠ 0 ∧ b ≠ 0 then h a b else 0) := by |
| by_cases ha : a = 0 |
| · subst a |
| simp [hleft] |
| · by_cases hb : b = 0 |
| · subst b |
| simp [hright] |
| · simp [ha, hb] |
| calc |
| fullAverage p q h = |
| fullAverage p q (fun a b => if a = 0 ∨ b = 0 then 1 else 0) + |
| nonzeroFlow p q h := by |
| unfold fullAverage nonzeroFlow |
| simp only [← Finset.sum_add_distrib] |
| apply Finset.sum_congr rfl |
| intro a _ |
| apply Finset.sum_congr rfl |
| intro b _ |
| simpa only [mul_add] using congrArg (fun z : ℝ => p a * q b * z) (hpoint a b) |
| _ = _ := by rw [boundary_mass p q hp hq] |
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| def conditionedRetention (p : A → ℝ) (q : B → ℝ) (h : A → B → ℝ) : ℝ := |
| nonzeroFlow p q h / (1 - zeroMass p q) |
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| theorem fullAverage_eq_mixture (p : A → ℝ) (q : B → ℝ) |
| (hp : IsProbability p) (hq : IsProbability q) (h : A → B → ℝ) |
| (hleft : ∀ b, h 0 b = 1) (hright : ∀ a, h a 0 = 1) |
| (hzero : zeroMass p q < 1) : |
| fullAverage p q h = zeroMass p q + |
| (1 - zeroMass p q) * conditionedRetention p q h := by |
| rw [fullAverage_eq_zeroMass_add_nonzeroFlow p q hp hq h hleft hright] |
| unfold conditionedRetention |
| have hne : 1 - zeroMass p q ≠ 0 := by linarith |
| field_simp |
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| / |
| inequality under an assumed mixture identity. -/ |
| theorem conditionedRetention_ge_half (p : A → ℝ) (q : B → ℝ) |
| (hp : IsProbability p) (hq : IsProbability q) (h : A → B → ℝ) |
| (hleft : ∀ b, h 0 b = 1) (hright : ∀ a, h a 0 = 1) |
| {η : ℝ} (hη : 0 < η) (hη1 : η < 1) |
| (hzero : 0 ≤ zeroMass p q) (hzeroη : zeroMass p q ≤ η / 2) |
| (haccept : η ≤ fullAverage p q h) : η / 2 ≤ conditionedRetention p q h := by |
| apply EqualityFiber.nonzero_retention_ge_half hη hη1 hzero hzeroη |
| rw [← fullAverage_eq_mixture p q hp hq h hleft hright (by linarith)] |
| exact haccept |
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| omit [Fintype A] [Fintype B] in |
| / |
| spaces. The powers are supplied by the cardinalities, so the formula also |
| keeps track of the overlap when both factors are zero. -/ |
| theorem zeroMass_binary_uniform (p : A → ℝ) (q : B → ℝ) (ell m : ℕ) |
| (hp0 : p 0 = ((2 : ℝ) ^ ell)⁻¹) (hq0 : q 0 = ((2 : ℝ) ^ m)⁻¹) : |
| zeroMass p q = ((2 : ℝ) ^ ell)⁻¹ + ((2 : ℝ) ^ m)⁻¹ - ((2 : ℝ) ^ (ell + m))⁻¹ := by |
| unfold zeroMass |
| rw [hp0, hq0, pow_add, mul_inv] |
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| end |
| end UniqueGamesTheorem.Inverse.FactorSampling |
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| end |
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| end OAI |
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