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67aee1d | 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 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 | import OAI.Analysis.FreeEntropy.HeatAnalysis
namespace OAI
universe u1
noncomputable section
open scoped ENNReal NNReal Topology
open MeasureTheory Filter Set Finset
namespace FiniteEntropySeparation
theorem signed_lintegral_toEReal {α : Type u1} [MeasurableSpace α] {μ : Measure α}
{b f : α → ℝ≥0∞} {r : α → ℝ}
(hb : ∀ᵐ x ∂μ, b x ≠ ⊤) (hf : ∀ᵐ x ∂μ, f x ≠ ⊤)
(hr : ∀ᵐ x ∂μ, r x = (b x).toReal - (f x).toReal) (hi : Integrable r μ) :
(∫⁻ x, b x - f x ∂μ).toEReal - (∫⁻ x, f x - b x ∂μ).toEReal =
((∫ x, r x ∂μ : ℝ) : EReal) := by
have hp : ∫⁻ x, b x - f x ∂μ = ∫⁻ x, ENNReal.ofReal (r x) ∂μ := by
apply lintegral_congr_ae
filter_upwards [hb,hf,hr] with x hb hf hr
rw [hr, ENNReal.ofReal_sub _ ENNReal.toReal_nonneg, ENNReal.ofReal_toReal hb,
ENNReal.ofReal_toReal hf]
have hn : ∫⁻ x, f x - b x ∂μ = ∫⁻ x, ENNReal.ofReal (-r x) ∂μ := by
apply lintegral_congr_ae
filter_upwards [hb,hf,hr] with x hb hf hr
rw [hr, neg_sub, ENNReal.ofReal_sub _ ENNReal.toReal_nonneg, ENNReal.ofReal_toReal hb,
ENNReal.ofReal_toReal hf]
have hnf : (∫⁻ x, ENNReal.ofReal (-r x) ∂μ) < ∞ := by
simpa only [Pi.neg_apply] using lintegral_ofReal_finite hi.neg
rw [hp,hn, ← EReal.coe_ennreal_toReal (lintegral_ofReal_finite hi).ne,
← EReal.coe_ennreal_toReal hnf.ne,
integral_eq_lintegral_pos_part_sub_lintegral_neg_part hi]
norm_cast
namespace Model
variable (p : ℕ) [NeZero p]
def gaussianEntropy (a : ℝ) : ℝ :=
(p+1 : ℕ)/2 * Real.log (2*Real.pi*Real.exp 1) +
(1/2 : ℝ) * ∑ i : Fin (p+1), Real.log (a + variance p i)
theorem chiStar_integral (hp : 2 ≤ p) (a : ℝ) (ha : 0 < a) :
chiStar (τ p) (X p a) (auxiliaryNoise p) =
(((p+1 : ℕ)/2 * Real.log (2*Real.pi*Real.exp 1) +
(1/2 : ℝ) * ∫ s in Ioi (0 : ℝ), (p+1 : ℕ)/(1+s) - heatFisherReal p (a+s) : ℝ) : EReal) := by
let b : ℝ → ℝ≥0∞ := fun s => ENNReal.ofReal ((p+1 : ℕ)/(1+s))
let f : ℝ → ℝ≥0∞ := fun s => freeFisher (τ p) (doublyNoised p a s)
let r : ℝ → ℝ := fun s => (p+1 : ℕ)/(1+s) - heatFisherReal p (a+s)
have hb : ∀ᵐ s ∂volume.restrict (Ioi (0 : ℝ)), b s ≠ ⊤ := Eventually.of_forall (fun _ => ENNReal.ofReal_ne_top)
have hf : ∀ᵐ s ∂volume.restrict (Ioi (0 : ℝ)), f s ≠ ⊤ := by
filter_upwards [ae_restrict_mem measurableSet_Ioi] with s hs
exact ne_top_of_le_ne_top ENNReal.ofReal_ne_top
(doublyNoised_fisher_le p a s ha.le hs.le (by linarith [show 0<s from hs]))
have hr : ∀ᵐ s ∂volume.restrict (Ioi (0 : ℝ)), r s = (b s).toReal - (f s).toReal := by
filter_upwards [ae_restrict_mem measurableSet_Ioi] with s hs
have hu : 0 < a+s := by linarith [show 0<s from hs]
have hs0 : 0 < s := hs
dsimp [r,b,f,heatFisherReal]
rw [doublyNoised_fisher_eq p a s ha.le hs.le hu, ENNReal.toReal_ofReal (by positivity)]
have he := signed_lintegral_toEReal hb hf hr (entropyIntegrand_integrable p hp a ha)
unfold chiStar
change (_ : EReal) + (1/2 : EReal) * (∫⁻ s in Ioi (0:ℝ), b s - f s).toEReal -
(1/2 : EReal) * (∫⁻ s in Ioi (0:ℝ), f s - b s).toEReal = _
have ht := entropy_parts_finite (f := f) (n := (p+1 : ℕ)) ha (by linarith : 0 < a+4)
(fun s hs => by simpa [f,add_assoc,add_comm,add_left_comm] using
doublyNoised_fisher_lower p hp a s ha.le hs.le (by linarith))
(fun s hs => doublyNoised_fisher_le p a s ha.le hs.le (by linarith))
rw [← EReal.coe_ennreal_toReal ht.1.ne, ← EReal.coe_ennreal_toReal ht.2.ne] at he ⊢
have he' : (∫⁻ s in Ioi (0:ℝ), b s - f s).toReal -
(∫⁻ s in Ioi (0:ℝ), f s - b s).toReal = ∫ s in Ioi (0:ℝ), r s := by
exact_mod_cast he
change (((p+1 : ℕ)/2 * Real.log (2*Real.pi*Real.exp 1) +
(1/2 : ℝ)*(∫⁻ s in Ioi (0:ℝ), b s-f s).toReal -
(1/2 : ℝ)*(∫⁻ s in Ioi (0:ℝ), f s-b s).toReal : ℝ) : EReal) = _
congr 1
change _ = (p+1 : ℕ)/2 * Real.log (2*Real.pi*Real.exp 1) +
(1/2 : ℝ)*∫ s in Ioi (0:ℝ), r s
linarith [he']
theorem covarianceCorrection_integrable (a : ℝ) (ha : 0 < a) :
IntegrableOn (fun s => ∑ i : Fin (p+1), reciprocalDifference (a+variance p i) s) (Ioi 0) :=
integrable_finsetSum _ (fun i _ => reciprocalDifference_integrable (by linarith [variance_nonneg p i]))
theorem covarianceCorrection_integral (a : ℝ) (ha : 0 < a) :
(∫ s in Ioi (0 : ℝ), ∑ i : Fin (p+1), reciprocalDifference (a+variance p i) s) =
∑ i : Fin (p+1), Real.log (a+variance p i) := by
rw [integral_finsetSum _ (fun i _ => reciprocalDifference_integrable (by linarith [variance_nonneg p i]))]
apply sum_congr rfl
intro i _
exact reciprocalDifference_integral (by linarith [variance_nonneg p i])
theorem covarianceCorrection_identity (a s : ℝ) :
(∑ i : Fin (p+1), reciprocalDifference (a+variance p i) s) -
((p+1 : ℕ)/(1+s) - heatFisherReal p (a+s)) = fisherDeficit p (a+s) := by
simp only [reciprocalDifference, sum_sub_distrib, sum_const, card_univ, Fintype.card_fin, nsmul_eq_mul]
simp only [fisherDeficit, covarianceFisher, heatFisherReal, one_div]
have he : (∑ i : Fin (p+1), (a+variance p i+s)⁻¹) = ∑ i : Fin (p+1), (variance p i+(a+s))⁻¹ := by
apply sum_congr rfl; intro i _; congr 1; ring
rw [he]
simp only [div_eq_mul_inv]
ring
theorem fisherDeficit_shift_integrable (hp : 2 ≤ p) (a : ℝ) (ha : 0 < a) :
IntegrableOn (fun s => fisherDeficit p (a+s)) (Ioi 0) := by
have hh := (covarianceCorrection_integrable p a ha).sub (entropyIntegrand_integrable p hp a ha)
change IntegrableOn (fun s => (∑ i : Fin (p+1), reciprocalDifference (a+variance p i) s) -
((p+1 : ℕ)/(1+s) - heatFisherReal p (a+s))) (Ioi 0) at hh
simpa only [covarianceCorrection_identity] using hh
theorem chiStar_covariance_formula (hp : 2 ≤ p) (a : ℝ) (ha : 0 < a) :
chiStar (τ p) (X p a) (auxiliaryNoise p) =
((gaussianEntropy p a - (1/2 : ℝ) * ∫ s in Ioi (0:ℝ), fisherDeficit p (a+s) : ℝ) : EReal) := by
rw [chiStar_integral p hp a ha]
congr 1
have he : (∫ s in Ioi (0:ℝ), fisherDeficit p (a+s)) =
(∑ i : Fin (p+1), Real.log (a+variance p i)) -
(∫ s in Ioi (0:ℝ), (p+1 : ℕ)/(1+s) - heatFisherReal p (a+s)) := by
simp_rw [← covarianceCorrection_identity p a]
rw [integral_sub (covarianceCorrection_integrable p a ha) (entropyIntegrand_integrable p hp a ha),
covarianceCorrection_integral p a ha]
rw [he]
unfold gaussianEntropy
ring
end Model
end FiniteEntropySeparation
end
end OAI
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