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c4cdbbd | 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 | import OAI.Probability.InvariantIsing.Core.Model
namespace OAI
/-! Finite entropy and the log-sum variational bound used to project
canonical spins onto an attainable constrained block. Zero trial masses
are allowed and contribute zero. -/
noncomputable section
open scoped BigOperators
namespace InvariantIsing
def finiteShannonEntropy {X : Type*} [Fintype X] (p : X → ℝ) : ℝ :=
-∑ x, p x * Real.log (p x)
lemma finite_relativeEntropy_nonneg_of_sum_le {X : Type*} [Fintype X] (p q : X → ℝ)
(hp : ∀ x, 0 ≤ p x) (hpSum : ∑ x, p x = 1)
(hq : ∀ x, 0 < q x) (hqSum : ∑ x, q x ≤ 1) :
0 ≤ ∑ x, p x * Real.log (p x / q x) := by
have hi (x : X) : p x - q x ≤ p x * Real.log (p x / q x) := by
by_cases hx : p x = 0
· simp only [hx, zero_sub, zero_mul]
exact neg_nonpos.mpr (hq x).le
have hxp : 0 < p x := lt_of_le_of_ne (hp x) (Ne.symm hx)
have hl := mul_le_mul_of_nonneg_left
(Real.log_le_sub_one_of_pos (div_pos (hq x) hxp)) (hp x)
have he : p x * (q x / p x - 1) = q x - p x := by field_simp
rw [he, Real.log_div (hq x).ne' hx] at hl
rw [Real.log_div hx (hq x).ne']
nlinarith
have hs := Finset.sum_le_sum (s := Finset.univ) (fun x _ => hi x)
rw [Finset.sum_sub_distrib, hpSum] at hs
linarith
lemma finite_relativeEntropy_nonneg {X : Type*} [Fintype X] (p q : X → ℝ)
(hp : ∀ x, 0 ≤ p x) (hpSum : ∑ x, p x = 1)
(hq : ∀ x, 0 < q x) (hqSum : ∑ x, q x = 1) :
0 ≤ ∑ x, p x * Real.log (p x / q x) :=
finite_relativeEntropy_nonneg_of_sum_le p q hp hpSum hq hqSum.le
def finiteCanonicalWeights {X : Type*} [Fintype X] (H : X → ℝ) (x : X) : ℝ :=
Real.exp (H x) / ∑ y, Real.exp (H y)
lemma finiteCanonicalWeights_pos {X : Type*} [Fintype X] [Nonempty X]
(H : X → ℝ) (x : X) : 0 < finiteCanonicalWeights H x :=
div_pos (Real.exp_pos _) (sum_exp_pos H)
lemma finiteCanonicalWeights_sum {X : Type*} [Fintype X] [Nonempty X]
(H : X → ℝ) : ∑ x, finiteCanonicalWeights H x = 1 := by
simp only [finiteCanonicalWeights, ← Finset.sum_div]
exact div_self (sum_exp_pos H).ne'
lemma log_sum_exp_ge_trial {X : Type*} [Fintype X] [Nonempty X]
(H p : X → ℝ) (hp : ∀ x, 0 ≤ p x) (hpSum : ∑ x, p x = 1) :
(∑ x, p x * H x) + finiteShannonEntropy p ≤ Real.log (∑ x, Real.exp (H x)) := by
have hk := finite_relativeEntropy_nonneg p (finiteCanonicalWeights H) hp hpSum
(finiteCanonicalWeights_pos H) (finiteCanonicalWeights_sum H)
have hi (x : X) : p x * Real.log (p x / finiteCanonicalWeights H x) =
p x * Real.log (p x) - p x * H x + p x * Real.log (∑ y, Real.exp (H y)) := by
by_cases hx : p x = 0
· simp [hx]
· rw [Real.log_div hx (finiteCanonicalWeights_pos H x).ne', finiteCanonicalWeights,
Real.log_div (Real.exp_ne_zero _) (sum_exp_pos H).ne', Real.log_exp]
ring
simp_rw [hi] at hk
rw [Finset.sum_add_distrib, Finset.sum_sub_distrib, ← Finset.sum_mul, hpSum, one_mul] at hk
unfold finiteShannonEntropy
linarith
lemma log_sum_exp_eq_canonical {X : Type*} [Fintype X] [Nonempty X] (H : X → ℝ) :
Real.log (∑ x, Real.exp (H x)) =
(∑ x, finiteCanonicalWeights H x * H x) + finiteShannonEntropy (finiteCanonicalWeights H) := by
have he (x : X) : finiteCanonicalWeights H x * Real.log (finiteCanonicalWeights H x) =
finiteCanonicalWeights H x * H x -
finiteCanonicalWeights H x * Real.log (∑ y, Real.exp (H y)) := by
conv_lhs => arg 2; rw [finiteCanonicalWeights, Real.log_div (Real.exp_ne_zero _)
(sum_exp_pos H).ne', Real.log_exp]
ring
unfold finiteShannonEntropy
simp_rw [he]
rw [Finset.sum_sub_distrib, ← Finset.sum_mul, finiteCanonicalWeights_sum, one_mul]
ring
end InvariantIsing
end
end OAI
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