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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 | import Mathlib.InformationTheory.KullbackLeibler.Basic
import Mathlib.Probability.ConditionalProbability
namespace OAI
/-! Exact relative entropy of conditioning on a nonnull event. -/
noncomputable section
open MeasureTheory ProbabilityTheory InformationTheory Set
namespace InvariantIsing
theorem conditioned_measure_entropy {X : Type*} [MeasurableSpace X]
(μ : Measure X) [IsProbabilityMeasure μ] (S : Set X) (hS : MeasurableSet S) (hpos : μ S ≠ 0) :
klDiv (cond μ S) μ ≠ ⊤ ∧ (klDiv (cond μ S) μ).toReal = -Real.log (μ.real S) := by
let ν := cond μ S
let : IsProbabilityMeasure ν := cond_isProbabilityMeasure hpos
have hν : ν ≪ μ := cond_absolutelyContinuous
have hνr : ν ≪ μ.restrict S := Measure.smul_absolutelyContinuous
have hs := hνr.ae_le (llr_smul_left (show μ.restrict S ≪ μ from Measure.restrict_le_self.absolutelyContinuous)
(μ S)⁻¹ (ENNReal.inv_ne_zero.mpr (measure_ne_top μ S)) (ENNReal.inv_ne_top.mpr hpos))
have hr := hν.ae_le (Measure.rnDeriv_restrict_self μ hS)
have hl : llr ν μ =ᵐ[ν] fun _ => -Real.log (μ.real S) := by
filter_upwards [hs, hr, ae_cond_mem hS (μ := μ)] with x hx hx' hxS
change llr ((μ S)⁻¹ • μ.restrict S) μ x = _
rw [hx]
change Real.log ((μ.restrict S).rnDeriv μ x).toReal + Real.log ((μ S)⁻¹).toReal = _
rw [hx', Set.indicator_of_mem hxS]
simp only [Pi.one_apply, ENNReal.toReal_one, Real.log_one, zero_add,
ENNReal.toReal_inv, Real.log_inv, measureReal_def]
have hi : Integrable (llr ν μ) ν := (integrable_const _).congr hl.symm
refine ⟨klDiv_ne_top hν hi, ?_⟩
rw [toReal_klDiv hν hi, integral_congr_ae hl]
simp only [integral_const, measureReal_def, measure_univ, ENNReal.toReal_one,
one_smul, add_sub_cancel_right]
end InvariantIsing
end
end OAI
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