| 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 |
|
|
