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openAI-math-dataset / lean /OAI /Computability /VertexCover /Information /LogSum.lean
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Add lean source and binary files (part 64)
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import Mathlib
import OAI.Computability.VertexCover.Information.Finite
namespace OAI
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namespace UniqueGames.Foundations.Information
open scoped BigOperators
variable {α : Type*} [Fintype α]
theorem normalize_isProbability (b : α → ℝ) (hb : ∀ a, 0 ≤ b a)
{B : ℝ} (hB : 0 < B) (hmass : ∑ a, b a = B) :
IsProbability (fun a => b a / B) := by
constructor
· intro a
exact div_nonneg (hb a) hB.le
· simp only [div_eq_mul_inv, ← Finset.sum_mul, hmass, mul_inv_cancel₀ hB.ne']
theorem relativeEntropy_normalize_right (p b : α → ℝ) (hp : IsProbability p)
(hs : SupportedBy p b) {B : ℝ} (hB : 0 < B) :
relativeEntropy p (fun a => b a / B) = relativeEntropy p b + Real.log B := by
have hpoint : ∀ a, p a * Real.log (p a / (b a / B)) =
p a * Real.log (p a / b a) + p a * Real.log B := by
intro a
by_cases hpa : p a = 0
· simp [hpa]
have hbne := hs a hpa
rw [Real.log_div hpa (div_ne_zero hbne hB.ne'), Real.log_div hbne hB.ne',
Real.log_div hpa hbne]
ring
simp only [relativeEntropy, hpoint, Finset.sum_add_distrib, ← Finset.sum_mul, hp.2, one_mul]
theorem neg_relativeEntropy_le_log_mass (p b : α → ℝ) (hp : IsProbability p)
(hb : ∀ a, 0 ≤ b a) (hs : SupportedBy p b) {B : ℝ} (hB : 0 < B)
(hmass : ∑ a, b a = B) : -relativeEntropy p b ≤ Real.log B := by
have hs' : SupportedBy p (fun a => b a / B) :=
fun a ha => div_ne_zero (hs a ha) hB.ne'
have hnonneg := relativeEntropy_nonneg p (fun a => b a / B) hp
(normalize_isProbability b hb hB hmass) hs'
rw [relativeEntropy_normalize_right p b hp hs hB] at hnonneg
linarith
theorem posterior_weighted_log_inverse_le (p b «c» : α → ℝ) (hp : IsProbability p)
(hb : ∀ a, 0 ≤ b a) {z B : ℝ} (hz : 0 < z) (hB : 0 < B)
(hmass : ∑ a, b a = B) (hpost : ∀ a, p a = b a * «c» a / z) :
(∑ a, p a * Real.log (1 / «c» a)) ≤ Real.log (B / z) := by
have hs : SupportedBy p b := by
intro a ha hba
apply ha
rw [hpost, hba, zero_mul, zero_div]
have hpoint : ∀ a, p a * Real.log (1 / «c» a) =
p a * Real.log (1 / z) - p a * Real.log (p a / b a) := by
intro a
by_cases hpa : p a = 0
· simp [hpa]
have hbne := hs a hpa
have hcne : «c» a ≠ 0 := by
intro hca
apply hpa
rw [hpost, hca, mul_zero, zero_div]
have hratio : p a / b a = «c» a / z := by
rw [hpost]
field_simp [hbne, hz.ne']
rw [hratio, Real.log_div hcne hz.ne', Real.log_div one_ne_zero hcne,
Real.log_div one_ne_zero hz.ne', Real.log_one]
ring
have hid : (∑ a, p a * Real.log (1 / «c» a)) =
Real.log (1 / z) - relativeEntropy p b := by
simp only [hpoint, Finset.sum_sub_distrib, ← Finset.sum_mul, hp.2, one_mul,
relativeEntropy]
have hkl := neg_relativeEntropy_le_log_mass p b hp hb hs hB hmass
rw [hid, Real.log_div hB.ne' hz.ne', Real.log_div one_ne_zero hz.ne', Real.log_one]
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