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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 | import OAI.Analysis.FreeEntropy.EntropyDeficit
namespace OAI
noncomputable section
open MeasureTheory Set Filter Finset
namespace FiniteEntropySeparation.Model
variable (p : ℕ) [NeZero p]
theorem reciprocal_variance_intervalIntegrable (i : Fin (p+1)) (a b : ℝ)
(ha : 0 < a) (hab : a ≤ b) :
IntervalIntegrable (fun u => (variance p i + u)⁻¹) volume a b := by
apply ContinuousOn.intervalIntegrable
apply ContinuousOn.inv₀ (continuous_const.add continuous_id).continuousOn
intro u hu
rw [uIcc_of_le hab] at hu
change variance p i + u ≠ 0
exact ne_of_gt (by linarith [variance_nonneg p i, hu.1])
theorem covarianceFisher_intervalIntegrable (a b : ℝ) (ha : 0 < a) (hab : a ≤ b) :
IntervalIntegrable (covarianceFisher p) volume a b := by
change IntervalIntegrable (fun u => ∑ i : Fin (p+1), (variance p i + u)⁻¹) volume a b
simpa only [Finset.sum_fn] using (IntervalIntegrable.sum univ (fun i _ => reciprocal_variance_intervalIntegrable p i a b ha hab))
theorem covarianceFisher_integral (a b : ℝ) (ha : 0 < a) (hab : a ≤ b) :
(∫ u in a..b, covarianceFisher p u) =
∑ i : Fin (p+1), (Real.log (b + variance p i) - Real.log (a + variance p i)) := by
unfold covarianceFisher
rw [intervalIntegral.integral_finsetSum (fun i _ => reciprocal_variance_intervalIntegrable p i a b ha hab)]
apply sum_congr rfl
intro i _
rw [intervalIntegral.integral_comp_add_left (fun u : ℝ => u⁻¹), integral_inv_of_pos
(by linarith [variance_nonneg p i]) (by linarith [variance_nonneg p i]),
Real.log_div (by linarith [variance_nonneg p i] : variance p i+b ≠ 0)
(by linarith [variance_nonneg p i] : variance p i+a ≠ 0)]
simp only [add_comm]
theorem gaussianEntropy_flow (a b : ℝ) (ha : 0 < a) (hab : a ≤ b) :
gaussianEntropy p b - gaussianEntropy p a =
(1/2 : ℝ) * ∫ u in a..b, covarianceFisher p u := by
rw [covarianceFisher_integral p a b ha hab, sum_sub_distrib]
unfold gaussianEntropy
ring
theorem fisherDeficit_intervalIntegrable (hp : 2 ≤ p) (a b : ℝ) (ha : 0 < a) (hab : a ≤ b) :
IntervalIntegrable (fisherDeficit p) volume a b := by
rw [intervalIntegrable_iff_integrableOn_Ioc_of_le hab]
exact (fisherDeficit_tail_integrable p hp a ha).mono_set (fun u hu => hu.1)
theorem heatFisherReal_intervalIntegrable (hp : 2 ≤ p) (a b : ℝ) (ha : 0 < a) (hab : a ≤ b) :
IntervalIntegrable (heatFisherReal p) volume a b := by
have hh := (fisherDeficit_intervalIntegrable p hp a b ha hab).add
(covarianceFisher_intervalIntegrable p a b ha hab)
change IntervalIntegrable (fun x => (freeFisher (τ p) (X p x)).toReal) volume a b
simpa only [fisherDeficit, sub_add_cancel] using hh
theorem chiStar_flow (hp : 2 ≤ p) (a b : ℝ) (ha : 0 < a) (hab : a ≤ b) :
chiStar (τ p) (X p b) (auxiliaryNoise p) - chiStar (τ p) (X p a) (auxiliaryNoise p) =
(((1/2 : ℝ) * ∫ u in a..b, heatFisherReal p u : ℝ) : EReal) := by
rw [chiStar_tail_formula p hp a ha, chiStar_tail_formula p hp b (lt_of_lt_of_le ha hab),
← EReal.coe_sub]
congr 1
have hd := intervalIntegral.integral_Ioi_sub_Ioi (fisherDeficit_tail_integrable p hp a ha) hab
have hg := gaussianEntropy_flow p a b ha hab
have hh : (∫ u in a..b, heatFisherReal p u) =
(∫ u in a..b, fisherDeficit p u) + (∫ u in a..b, covarianceFisher p u) := by
rw [← intervalIntegral.integral_add (fisherDeficit_intervalIntegrable p hp a b ha hab)
(covarianceFisher_intervalIntegrable p a b ha hab)]
apply intervalIntegral.integral_congr
intro u hu
simp only [fisherDeficit, heatFisherReal, sub_add_cancel]
rw [hh]
linarith
end FiniteEntropySeparation.Model
end
end OAI
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