| import OAI.Probability.FreeIsing.Estimates.BalancedLatentMean |
| import OAI.Probability.FreeIsing.Trees.NoisyCutLevelIndependence |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal |
|
|
| namespace Problem367.RootedBroadcast |
|
|
| local instance cutLevelConditionalMeanMeasurableSingletonIsingSpin : |
| MeasurableSingletonClass IsingSpin := ⟨fun _ => trivial⟩ |
| local instance cutLevelConditionalMeanIsProbabilityLabelMeasure (d : ℕ) : |
| IsProbabilityMeasure (labelMeasure d) := by |
| unfold labelMeasure |
| infer_instance |
|
|
| theorem cutSpins_coordinate_fair {d : ℕ} {theta : ℝ} |
| (h0 : -1 ≤ theta) (h1 : theta ≤ 1) (n : ℕ) (c : TreeShell d n) (a : IsingSpin) : |
| ((labelMeasure d).map (cutSpins theta n)) {s | s c = a} = (1 / 2 : ℝ≥0∞) := by |
| rw [Measure.map_apply (measurable_cutSpins d theta n) |
| (measurableSet_eq_fun (measurable_pi_apply c) measurable_const)] |
| have h := (freeIsingMeasure_hasFreeIsingLaw d h0 h1).marginal c.1 a |
| rw [freeIsingMeasure, Measure.map_apply (measurable_factor d theta) |
| (measurableSet_eq_fun (measurable_pi_apply c.1) measurable_const)] at h |
| change (labelMeasure d) {U | factor theta U c.1 = a} = _ |
| change (labelMeasure d) {U | factor theta U c.1 = a} = _ at h |
| simpa using h |
|
|
| theorem condExp_noisy_factor_cone_le_two_integral {d : ℕ} {theta : ℝ} {A : Type*} |
| [MeasurableSpace A] [StandardBorelSpace A] |
| (h0 : -1 ≤ theta) (h1 : theta ≤ 1) (n : ℕ) |
| (ν : TreeVertex d → Measure A) [∀ v, IsProbabilityMeasure (ν v)] |
| (c : TreeShell d n) |
| (F : (DescendantVertex c.1 → IsingSpin) × (DescendantVertex c.1 → A) → ℝ) |
| (hF : Measurable F) (hbound : ∀ p, 0 ≤ F p ∧ F p ≤ 1) : |
| ∀ᵐ z ∂(labelMeasure d).prod (Measure.infinitePi ν), |
| (((labelMeasure d).prod (Measure.infinitePi ν))[ |
| (fun z : (TreeVertex d → ℝ) × (TreeVertex d → A) => |
| F ((fun v => factor theta z.1 v.1), fun v => z.2 v.1)) | |
| (inferInstance : MeasurableSpace (TreeShell d n → IsingSpin)).comap |
| (fun z => cutSpins theta n z.1)]) z ≤ |
| 2 * ∫ w : (TreeVertex d → ℝ) × (TreeVertex d → A), |
| F ((fun v => factor theta w.1 v.1), fun v => w.2 v.1) |
| ∂(labelMeasure d).prod (Measure.infinitePi ν) := by |
| let S := TreeShell d n → IsingSpin |
| let R := (e : TreeShell d n) → |
| (StrictDescendantVertex e.1 → ℝ) × (DescendantVertex e.1 → A) |
| let rho := (labelMeasure d).map (cutSpins theta n) |
| let muR := Measure.infinitePi (fun e : TreeShell d n => |
| (Measure.infinitePi (fun _ : StrictDescendantVertex e.1 => uniform01)).prod |
| (Measure.infinitePi (fun v : DescendantVertex e.1 => ν v.1))) |
| let e : (TreeVertex d → ℝ) × (TreeVertex d → A) → S × R := |
| fun z => (cutSpins theta n z.1, |
| fun c => (cutLabels z.1 c, fun v => z.2 v.1)) |
| have he : MeasurePreserving e ((labelMeasure d).prod (Measure.infinitePi ν)) |
| (rho.prod muR) := |
| ⟨((measurable_cutSpins d theta n).comp measurable_fst).prodMk |
| (by unfold cutLabels; fun_prop), map_cutSpins_cutLabels_noises theta n ν⟩ |
| let G : IsingSpin × R → ℝ := fun p => |
| F (coneFactor theta c.1 (p.1, (p.2 c).1), (p.2 c).2) |
| have hG : Measurable G := |
| hF.comp (((measurable_coneFactor theta c.1).comp |
| (measurable_fst.prodMk |
| (measurable_fst.comp ((measurable_pi_apply c).comp measurable_snd)))).prodMk |
| (measurable_snd.comp ((measurable_pi_apply c).comp measurable_snd))) |
| have h := condExp_binary_channel_comp_le_two_integral |
| ((labelMeasure d).prod (Measure.infinitePi ν)) rho muR e he |
| (fun s : S => s c) (measurable_pi_apply c) |
| (cutSpins_coordinate_fair h0 h1 n c) G hG (fun p => hbound _) |
| have hfun : (fun z => G ((e z).1 c, (e z).2)) = |
| (fun z : (TreeVertex d → ℝ) × (TreeVertex d → A) => |
| F ((fun v => factor theta z.1 v.1), fun v => z.2 v.1)) := by |
| funext z |
| apply congrArg F |
| apply Prod.ext |
| · exact (factor_cone_representation theta c.1 z.1).symm |
| · rfl |
| rw [hfun] at h |
| exact h |
|
|
| end Problem367.RootedBroadcast |
|
|
| namespace Problem367 |
|
|
| local instance cutLevelConditionalMeanFreeIsingMeasurableSingletonIsingSpin : |
| MeasurableSingletonClass IsingSpin := ⟨fun _ => trivial⟩ |
|
|
| theorem freeIsingMeasure_condExp_cutLevel_observable_le_two_integral |
| {d : ℕ} {theta : ℝ} {A : Type*} |
| [MeasurableSpace A] [StandardBorelSpace A] |
| (h0 : -1 ≤ theta) (h1 : theta ≤ 1) (n : ℕ) |
| (ν : TreeVertex d → Measure A) [∀ v, IsProbabilityMeasure (ν v)] |
| (c : TreeShell d n) |
| (F : (RootedBroadcast.DescendantVertex c.1 → IsingSpin) × |
| (RootedBroadcast.DescendantVertex c.1 → A) → ℝ) |
| (hF : Measurable F) (hbound : ∀ p, 0 ≤ F p ∧ F p ≤ 1) : |
| ∀ᵐ z ∂(freeIsingMeasure d theta).prod (Measure.infinitePi ν), |
| (((freeIsingMeasure d theta).prod (Measure.infinitePi ν))[ |
| (fun z : (TreeVertex d → IsingSpin) × (TreeVertex d → A) => |
| F ((fun v => z.1 v.1), fun v => z.2 v.1)) | |
| (inferInstance : MeasurableSpace (TreeShell d n → IsingSpin)).comap |
| (fun z => fun e : TreeShell d n => z.1 e.1)]) z ≤ |
| 2 * ∫ w : (TreeVertex d → IsingSpin) × (TreeVertex d → A), |
| F ((fun v => w.1 v.1), fun v => w.2 v.1) |
| ∂(freeIsingMeasure d theta).prod (Measure.infinitePi ν) := by |
| let : IsProbabilityMeasure (labelMeasure d) := by |
| unfold labelMeasure |
| infer_instance |
| let P := (freeIsingMeasure d theta).prod (Measure.infinitePi ν) |
| let Q := (labelMeasure d).prod (Measure.infinitePi ν) |
| let e : (TreeVertex d → ℝ) × (TreeVertex d → A) → |
| (TreeVertex d → IsingSpin) × (TreeVertex d → A) := |
| Prod.map (RootedBroadcast.factor theta) id |
| let H : (TreeVertex d → IsingSpin) × (TreeVertex d → A) → ℝ := |
| fun z => F ((fun v => z.1 v.1), fun v => z.2 v.1) |
| let S : (TreeVertex d → IsingSpin) × (TreeVertex d → A) → |
| TreeShell d n → IsingSpin := fun z c => z.1 c.1 |
| have hS : Measurable S := by fun_prop |
| have hH : Measurable H := hF.comp (by fun_prop) |
| have hInt : Integrable H P := by |
| apply Integrable.of_bound hH.aestronglyMeasurable 1 |
| exact Filter.Eventually.of_forall (fun z => by |
| rw [Real.norm_eq_abs, abs_of_nonneg (hbound _).1] |
| exact (hbound _).2) |
| have hf : MeasurePreserving (RootedBroadcast.factor theta) |
| (labelMeasure d) (freeIsingMeasure d theta) := |
| ⟨RootedBroadcast.measurable_factor d theta, rfl⟩ |
| have he : MeasurePreserving e Q P := hf.prod (MeasurePreserving.id (Measure.infinitePi ν)) |
| have hpull := condExp_comp_measurePreserving Q P e he |
| ((inferInstance : MeasurableSpace (TreeShell d n → IsingSpin)).comap S) |
| hS.comap_le hInt |
| have hMean : (∫ x, H (e x) ∂Q) = ∫ z, H z ∂P := by |
| rw [← he.map_eq] |
| exact (integral_map he.measurable.aemeasurable hH.aestronglyMeasurable).symm |
| have hsource := RootedBroadcast.condExp_noisy_factor_cone_le_two_integral |
| h0 h1 n ν c F hF hbound |
| change ∀ᵐ z ∂P, (P[H | MeasurableSpace.comap S inferInstance]) z ≤ |
| 2 * ∫ w, H w ∂P |
| have hgoal : ∀ᵐ z ∂Q.map e, |
| (P[H | MeasurableSpace.comap S inferInstance]) z ≤ 2 * ∫ w, H w ∂P := by |
| apply (ae_map_iff he.measurable.aemeasurable |
| (measurableSet_le (stronglyMeasurable_condExp.mono hS.comap_le).measurable |
| measurable_const)).mpr |
| filter_upwards [hpull, hsource] with z hz hzbound |
| have hmeanbound : (Q[H ∘ e | (MeasurableSpace.comap S inferInstance).comap e]) z ≤ |
| 2 * ∫ x, H (e x) ∂Q := by |
| simp only [MeasurableSpace.comap_comp, Function.comp_def, S, H, e, |
| Prod.map_fst, Prod.map_snd, id_eq] |
| exact hzbound |
| rw [hz, hMean] at hmeanbound |
| exact hmeanbound |
| simpa only [he.map_eq] using hgoal |
|
|
| theorem HasFreeIsingLaw.condExp_cutLevel_observable_le_two_integral |
| {d : ℕ} {theta : ℝ} {A : Type*} |
| [MeasurableSpace A] [StandardBorelSpace A] |
| {μ : Measure (TreeVertex d → IsingSpin)} [IsFiniteMeasure μ] |
| (hμ : HasFreeIsingLaw theta μ) (h0 : -1 ≤ theta) (h1 : theta ≤ 1) (n : ℕ) |
| (ν : TreeVertex d → Measure A) [∀ v, IsProbabilityMeasure (ν v)] |
| (c : TreeShell d n) |
| (F : (RootedBroadcast.DescendantVertex c.1 → IsingSpin) × |
| (RootedBroadcast.DescendantVertex c.1 → A) → ℝ) |
| (hF : Measurable F) (hbound : ∀ p, 0 ≤ F p ∧ F p ≤ 1) : |
| ∀ᵐ z ∂μ.prod (Measure.infinitePi ν), |
| ((μ.prod (Measure.infinitePi ν))[ |
| (fun z : (TreeVertex d → IsingSpin) × (TreeVertex d → A) => |
| F ((fun v => z.1 v.1), fun v => z.2 v.1)) | |
| (inferInstance : MeasurableSpace (TreeShell d n → IsingSpin)).comap |
| (fun z => fun e : TreeShell d n => z.1 e.1)]) z ≤ |
| 2 * ∫ w : (TreeVertex d → IsingSpin) × (TreeVertex d → A), |
| F ((fun v => w.1 v.1), fun v => w.2 v.1) ∂μ.prod (Measure.infinitePi ν) := by |
| have heq := hμ.eq_freeIsingMeasure h0 h1 |
| subst μ |
| exact freeIsingMeasure_condExp_cutLevel_observable_le_two_integral h0 h1 n ν c F hF hbound |
|
|
| end Problem367 |
|
|
| end |
|
|
| end OAI |
|
|
