| import OAI.Probability.FreeIsing.Conditioning.ComponentInformation |
| import OAI.Probability.FreeIsing.Conditioning.Pullback |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory |
| open scoped NNReal |
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| namespace Problem367 |
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| section Transport |
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| variable {Ω Ξ : Type*} [mΩ : MeasurableSpace Ω] [mΞ : MeasurableSpace Ξ] |
| (μ : Measure Ω) (ν : Measure Ξ) [IsFiniteMeasure μ] [IsFiniteMeasure ν] |
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| theorem posterior_information_comp_measurePreserving |
| (e : Ω → Ξ) (he : MeasurePreserving e μ ν) |
| (m : MeasurableSpace Ξ) (hm : m ≤ mΞ) {f : Ξ → ℝ} (hf : Integrable f ν) : |
| (∫ ω, (condExp (m.comap e) μ (f ∘ e) ω) ^ 2 ∂μ) = |
| ∫ ξ, (condExp m ν f ξ) ^ 2 ∂ν := by |
| let : MeasurableSpace Ξ := mΞ |
| calc |
| _ = ∫ ω, (condExp m ν f (e ω)) ^ 2 ∂μ := by |
| apply integral_congr_ae |
| filter_upwards [condExp_comp_measurePreserving μ ν e he m hm hf] with ω hω |
| exact congrArg (fun z : ℝ => z ^ 2) hω |
| _ = _ := by |
| have hp : StronglyMeasurable (condExp m ν f) := stronglyMeasurable_condExp.mono hm |
| exact he.hasLaw.integral_comp (f := fun ξ => (condExp m ν f ξ) ^ 2) |
| (hp.pow 2).aestronglyMeasurable |
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| theorem posterior_information_le_of_measurePreserving |
| (e : Ω → Ξ) (he : MeasurePreserving e μ ν) |
| (m : MeasurableSpace Ξ) (hm : m ≤ mΞ) |
| (M : MeasurableSpace Ω) (hM : M ≤ mΩ) (hobs : m.comap e ≤ M) |
| {f : Ξ → ℝ} (hf : MemLp f 2 ν) : |
| (∫ ξ, (condExp m ν f ξ) ^ 2 ∂ν) ≤ |
| ∫ ω, (condExp M μ (f ∘ e) ω) ^ 2 ∂μ := by |
| let : MeasurableSpace Ω := mΩ |
| let : MeasurableSpace Ξ := mΞ |
| rw [← posterior_information_comp_measurePreserving μ ν e he m hm |
| (hf.integrable (by norm_num))] |
| exact posterior_information_mono hobs hM (hf.comp_measurePreserving he) |
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| end Transport |
|
|
| theorem HasFreeIsingLaw.component_critical_information_lower_transport |
| {d : ℕ} {theta : ℝ} {mu : Measure (TreeVertex d → IsingSpin)} |
| (hmu : HasFreeIsingLaw theta mu) (hlo : -1 ≤ theta) (hhi : theta ≤ 1) |
| (c : Fin d) (hcritical : ((d - 1 : ℕ) : ℝ) * theta ^ 2 = 1) |
| {t : ℝ≥0} (ht : 0 < t) |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsFiniteMeasure P] |
| (e : Ω → (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ)) |
| (he : MeasurePreserving e P (mu.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t))) |
| (M : MeasurableSpace Ω) (hM : M ≤ mΩ) |
| (hobs : (PlantedInformation.observationSigma (componentIndexVertex (c := c)) t).comap e ≤ M) : |
| min (Real.sqrt (t : ℝ)) 1 / 5 ≤ |
| ∫ ω, (condExp M P (fun η => spinSign ((e η).1 ⟨[c], List.IsChain.singleton c⟩)) ω) ^ 2 ∂P := by |
| let : MeasurableSpace Ω := mΩ |
| let := hmu.isProbabilityMeasure |
| apply (hmu.component_critical_information_lower hlo hhi c hcritical ht).trans |
| apply posterior_information_le_of_measurePreserving P |
| (mu.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t)) e he |
| (PlantedInformation.observationSigma (componentIndexVertex (c := c)) t) |
| (PlantedInformation.observationSigma_le _ _) M hM hobs |
| exact (PlantedInformation.spinSign_memLp mu ⟨[c], List.IsChain.singleton c⟩).comp_measurePreserving |
| measurePreserving_fst |
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| end Problem367 |
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| end |
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| end OAI |
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