| 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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|
