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e320e82 | 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 | import Mathlib
import OAI.Combinatorics.IndependentSets.Repetition.Finite
namespace OAI
namespace IndependentSetsGames.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 IndependentSetsGames.Foundations.Information
end OAI
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