| import Mathlib |
| import OAI.Computability.VertexCover.Information.Finite |
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| namespace OAI |
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| namespace UniqueGames.Foundations.Information |
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|
| open scoped BigOperators |
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| variable [Fintype α] [Fintype β] |
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| noncomputable def firstMarginal (p : α × β → ℝ) (a : α) : ℝ := ∑ b, p (a, b) |
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| noncomputable def secondMarginal (p : α × β → ℝ) (b : β) : ℝ := ∑ a, p (a, b) |
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| def product (p : α → ℝ) (q : β → ℝ) : α × β → ℝ := fun ab => p ab.1 * q ab.2 |
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| theorem firstMarginal_isProbability (p : α × β → ℝ) (hp : IsProbability p) : |
| IsProbability (firstMarginal p) := by |
| constructor |
| · intro a |
| exact Finset.sum_nonneg (fun b _ => hp.1 (a, b)) |
| · simpa only [firstMarginal, Fintype.sum_prod_type] using hp.2 |
| |
| theorem secondMarginal_isProbability (p : α × β → ℝ) (hp : IsProbability p) : |
| IsProbability (secondMarginal p) := by |
| constructor |
| · intro b |
| exact Finset.sum_nonneg (fun a _ => hp.1 (a, b)) |
| · change (∑ b, ∑ a, p (a, b)) = 1 |
| rw [Finset.sum_comm] |
| exact (firstMarginal_isProbability p hp).2 |
| |
| theorem product_isProbability (p : α → ℝ) (q : β → ℝ) |
| (hp : IsProbability p) (hq : IsProbability q) : IsProbability (product p q) := by |
| constructor |
| · intro ab |
| exact mul_nonneg (hp.1 ab.1) (hq.1 ab.2) |
| · simp only [product, Fintype.sum_prod_type, ← Finset.mul_sum, hq.2, mul_one, hp.2] |
| |
| theorem point_le_firstMarginal (p : α × β → ℝ) (hp : IsProbability p) (a : α) (b : β) : |
| p (a, b) ≤ firstMarginal p a := by |
| classical |
| exact Finset.single_le_sum (fun b _ => hp.1 (a, b)) (Finset.mem_univ b) |
|
|
| theorem point_le_secondMarginal (p : α × β → ℝ) (hp : IsProbability p) (a : α) (b : β) : |
| p (a, b) ≤ secondMarginal p b := by |
| classical |
| exact Finset.single_le_sum (fun a _ => hp.1 (a, b)) (Finset.mem_univ a) |
|
|
| theorem joint_supportedBy_product_marginals (p : α × β → ℝ) (hp : IsProbability p) : |
| SupportedBy p (product (firstMarginal p) (secondMarginal p)) := by |
| intro ab hpne |
| have hpos : 0 < p ab := lt_of_le_of_ne (hp.1 ab) (Ne.symm hpne) |
| exact mul_ne_zero |
| (ne_of_gt (hpos.trans_le (point_le_firstMarginal p hp ab.1 ab.2))) |
| (ne_of_gt (hpos.trans_le (point_le_secondMarginal p hp ab.1 ab.2))) |
|
|
| theorem relativeEntropy_product_reference (p : α × β → ℝ) (u : α → ℝ) (v : β → ℝ) |
| (hp : IsProbability p) (_hu : IsProbability u) (_hv : IsProbability v) |
| (hsu : SupportedBy (firstMarginal p) u) (hsv : SupportedBy (secondMarginal p) v) : |
| relativeEntropy p (product u v) = |
| relativeEntropy p (product (firstMarginal p) (secondMarginal p)) + |
| relativeEntropy (firstMarginal p) u + relativeEntropy (secondMarginal p) v := by |
| have hpoint : ∀ ab : α × β, p ab * Real.log (p ab / product u v ab) = |
| p ab * Real.log (p ab / product (firstMarginal p) (secondMarginal p) ab) + |
| p ab * Real.log (firstMarginal p ab.1 / u ab.1) + |
| p ab * Real.log (secondMarginal p ab.2 / v ab.2) := by |
| intro ab |
| by_cases hzero : p ab = 0 |
| · simp [hzero] |
| have hpos : 0 < p ab := lt_of_le_of_ne (hp.1 ab) (Ne.symm hzero) |
| have hleft := ne_of_gt (hpos.trans_le (point_le_firstMarginal p hp ab.1 ab.2)) |
| have hright := ne_of_gt (hpos.trans_le (point_le_secondMarginal p hp ab.1 ab.2)) |
| have hune := hsu ab.1 hleft |
| have hvne := hsv ab.2 hright |
| simp only [product, Real.log_div hzero (mul_ne_zero hune hvne), |
| Real.log_div hzero (mul_ne_zero hleft hright), Real.log_mul hune hvne, |
| Real.log_mul hleft hright, Real.log_div hleft hune, Real.log_div hright hvne] |
| ring |
| have hleftSum : (∑ ab : α × β, p ab * Real.log (firstMarginal p ab.1 / u ab.1)) = |
| relativeEntropy (firstMarginal p) u := by |
| simp only [Fintype.sum_prod_type, ← Finset.sum_mul, relativeEntropy, firstMarginal] |
| have hrightSum : (∑ ab : α × β, p ab * Real.log (secondMarginal p ab.2 / v ab.2)) = |
| relativeEntropy (secondMarginal p) v := by |
| rw [Fintype.sum_prod_type, Finset.sum_comm] |
| simp only [← Finset.sum_mul, relativeEntropy, secondMarginal] |
| unfold relativeEntropy |
| simp_rw [hpoint] |
| rw [Finset.sum_add_distrib, Finset.sum_add_distrib, hleftSum, hrightSum] |
| rfl |
|
|
| theorem marginal_relativeEntropy_sum_le (p : α × β → ℝ) (u : α → ℝ) (v : β → ℝ) |
| (hp : IsProbability p) (hu : IsProbability u) (hv : IsProbability v) |
| (hsu : SupportedBy (firstMarginal p) u) (hsv : SupportedBy (secondMarginal p) v) : |
| relativeEntropy (firstMarginal p) u + relativeEntropy (secondMarginal p) v ≤ |
| relativeEntropy p (product u v) := by |
| rw [relativeEntropy_product_reference p u v hp hu hv hsu hsv] |
| have hn := relativeEntropy_nonneg p (product (firstMarginal p) (secondMarginal p)) hp |
| (product_isProbability _ _ (firstMarginal_isProbability p hp) |
| (secondMarginal_isProbability p hp)) (joint_supportedBy_product_marginals p hp) |
| linarith |
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| end UniqueGames.Foundations.Information |
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| end |
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| end OAI |
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