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a89d282 | 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 | import OAI.Probability.InvariantIsing.Core.TiltedRelativeEntropy
import Mathlib.Probability.Moments.SubGaussian
namespace OAI
/-! Relative entropy controls the expectation of a canonical
sub-Gaussian observable. This avoids a loss from the ambient dimension. -/
noncomputable section
open MeasureTheory ProbabilityTheory InformationTheory
open scoped NNReal
namespace InvariantIsing
theorem entropy_variational_mean {X : Type*} [MeasurableSpace X]
(μ ν : Measure X) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
(hac : μ ≪ ν) (hllr : Integrable (llr μ ν) μ) (F : X → ℝ)
(hF : Integrable F μ) (he : Integrable (fun x => Real.exp (F x)) ν) :
(∫ x, F x ∂μ) ≤ (klDiv μ ν).toReal + Real.log (∫ x, Real.exp (F x) ∂ν) := by
let π := ν.tilted F
let : IsProbabilityMeasure π := isProbabilityMeasure_tilted he
have hac' : μ ≪ π := hac.trans (absolutelyContinuous_tilted he)
have hi : Integrable (llr μ π) μ := integrable_llr_tilted_right hac hF hllr he
have hn := integral_llr_add_sub_measure_univ_nonneg hac' hi
have hE := integral_llr_tilted_right hac hF he hllr
have hK := toReal_klDiv hac hllr
simp only [measureReal_def, measure_univ, ENNReal.toReal_one, add_sub_cancel_right] at hn hK
change 0 ≤ ∫ x, llr μ (ν.tilted F) x ∂μ at hn
linarith
theorem entropy_subgaussian_mean {X : Type*} [MeasurableSpace X]
(μ ν : Measure X) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
(hac : μ ≪ ν) (hllr : Integrable (llr μ ν) μ) (Y : X → ℝ) {c : ℝ≥0}
(hY : Integrable Y μ) (hG : HasSubgaussianMGF Y c ν) {t : ℝ} (ht : 0 < t) :
|∫ x, Y x ∂μ| ≤ (klDiv μ ν).toReal / t + c * t / 2 := by
have hp := entropy_variational_mean μ ν hac hllr (fun x => t * Y x)
(hY.const_mul t) (hG.integrable_exp_mul t)
have hn := entropy_variational_mean μ ν hac hllr (fun x => -t * Y x)
(hY.const_mul (-t)) (hG.integrable_exp_mul (-t))
have hgp := hG.cgf_le t
have hgn := hG.cgf_le (-t)
change Real.log (∫ x, Real.exp (t * Y x) ∂ν) ≤ _ at hgp
change Real.log (∫ x, Real.exp (-t * Y x) ∂ν) ≤ _ at hgn
rw [integral_const_mul] at hp hn
have hbound : t * |∫ x, Y x ∂μ| ≤ (klDiv μ ν).toReal + c * t ^ 2 / 2 := by
have h := (abs_le.mpr (show -( (klDiv μ ν).toReal + c * t ^ 2 / 2) ≤
t * (∫ x, Y x ∂μ) ∧ t * (∫ x, Y x ∂μ) ≤
(klDiv μ ν).toReal + c * t ^ 2 / 2 by constructor <;> nlinarith))
simpa only [abs_mul, abs_of_pos ht] using h
apply (mul_le_mul_iff_right₀ ht).mp
calc
_ = t * |∫ x, Y x ∂μ| := by ring
_ ≤ _ := hbound
_ = t * ((klDiv μ ν).toReal / t + c * t / 2) := by field_simp
end InvariantIsing
end
end OAI
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