| 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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|
