File size: 5,421 Bytes
92416a2 | 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 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 | import Mathlib
import OAI.Computability.VertexCover.Information.Finite
namespace OAI
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
namespace UniqueGames.Foundations.Information
open scoped BigOperators
variable {α β : Type*} [Fintype α] [Fintype β]
noncomputable def firstMarginal (p : α × β → ℝ) (a : α) : ℝ := ∑ b, p (a, b)
noncomputable def secondMarginal (p : α × β → ℝ) (b : β) : ℝ := ∑ a, p (a, b)
def product (p : α → ℝ) (q : β → ℝ) : α × β → ℝ := fun ab => p ab.1 * q ab.2
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
end UniqueGames.Foundations.Information
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end OAI
|