| import OAI.Probability.FreeIsing.Conditioning.ComponentInformationUpper |
| import OAI.Probability.FreeIsing.Conditioning.FiniteObservationInformation |
| import OAI.Probability.FreeIsing.Conditioning.NoisyBranchConditionalLaw |
| import OAI.Probability.FreeIsing.Critical.Information |
| import OAI.Probability.FreeIsing.Estimates.ThresholdArithmetic |
| import OAI.Probability.FreeIsing.Posterior.ComponentObservationDecomposition |
| import OAI.Probability.FreeIsing.Posterior.ComponentObservationIndependence |
| import OAI.Probability.FreeIsing.Posterior.ComponentObservationRegression |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory ProbabilityTheory |
| open scoped NNReal BigOperators |
|
|
| namespace Problem367 |
|
|
| theorem HasFreeIsingLaw.selected_child_parent_abs_le |
| {d : ℕ} {theta : ℝ} {A B : Type*} |
| [MeasurableSpace A] [StandardBorelSpace A] |
| [MeasurableSpace B] [StandardBorelSpace B] [Nonempty B] |
| {μ : Measure (TreeVertex d → IsingSpin)} (hμ : HasFreeIsingLaw theta μ) |
| (h0 : 0 ≤ theta) (h1 : theta ≤ 1) |
| (ν : Measure A) [IsProbabilityMeasure ν] |
| {F : IsingSpin → A → B} (hF : Measurable (Function.uncurry F)) (c a : Fin d) : |
| ∀ᵐ z ∂μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν)), |
| |selectedPosterior (μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν))) |
| F (childComponentIndexVertex c a) (RootedBroadcast.branchRoot c) z| ≤ theta := by |
| filter_upwards [hμ.selectedPosterior_parent_childComponent (by linarith) h1 ν hF c a, |
| selectedPosterior_abs_le_one (μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν))) |
| F (childComponentIndexVertex c a) (prefixFlip c (RootedBroadcast.branchRoot a))] with z hz hb |
| rw [hz, abs_mul, abs_of_nonneg h0] |
| simpa only [mul_one] using mul_le_mul_of_nonneg_left hb h0 |
|
|
| theorem HasFreeIsingLaw.selected_child_parent_secondMoment |
| {d : ℕ} {theta : ℝ} {A B : Type*} |
| [MeasurableSpace A] [StandardBorelSpace A] |
| [MeasurableSpace B] [StandardBorelSpace B] [Nonempty B] |
| {μ : Measure (TreeVertex d → IsingSpin)} (hμ : HasFreeIsingLaw theta μ) |
| (h0 : -1 ≤ theta) (h1 : theta ≤ 1) |
| (ν : Measure A) [IsProbabilityMeasure ν] |
| {F : IsingSpin → A → B} (hF : Measurable (Function.uncurry F)) (c a : Fin d) : |
| (∫ z, (selectedPosterior (μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν))) |
| F (childComponentIndexVertex c a) (RootedBroadcast.branchRoot c) z) ^ 2 |
| ∂μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν))) = |
| theta ^ 2 * ∫ z, (selectedPosterior (μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν))) |
| F (componentIndexVertex (c := c)) (RootedBroadcast.branchRoot c) z) ^ 2 |
| ∂μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν)) := by |
| have h := hμ.selectedPosterior_parent_childComponent h0 h1 ν hF c a |
| have heq := integral_congr_ae (h.fun_comp (fun x : ℝ => x ^ 2)) |
| simp only [Function.comp_apply] at heq |
| rw [heq] |
| simp only [mul_pow] |
| rw [integral_const_mul] |
| congr 1 |
| exact ((childComponentPosterior_identDistrib hμ ν hF c a).comp |
| (show Measurable (fun x : ℝ => x ^ 2) by fun_prop)).integral_eq |
|
|
| def gaussianComponentInformation {d : ℕ} |
| (μ : Measure (TreeVertex d → IsingSpin)) (t : ℝ≥0) (c : Fin d) : ℝ := |
| ∫ z, (selectedPosterior (μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t)) |
| (fun s x => (t : ℝ) * spinSign s + x) (componentIndexVertex (c := c)) |
| (RootedBroadcast.branchRoot c) z) ^ 2 |
| ∂μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t) |
|
|
| theorem HasFreeIsingLaw.gaussian_component_information_fusion |
| {d : ℕ} (hd : 3 ≤ d) {beta : ℝ} |
| {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (t : ℝ≥0) (c : Fin d) : |
| gaussianComponentInformation μ t c ≤ |
| ((d - 1 : ℕ) : ℝ) * (Real.tanh beta) ^ 2 * gaussianComponentInformation μ t c - |
| 2 * (1 - (Real.tanh beta) ^ 2) * (Real.tanh beta) ^ 4 * |
| (gaussianComponentInformation μ t c) ^ 2 + ((t : ℝ) + (t : ℝ) ^ 2) := by |
| classical |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure |
| let ρ := μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t) |
| let F : IsingSpin → ℝ → ℝ := fun s x => (t : ℝ) * spinSign s + x |
| let r := RootedBroadcast.branchRoot c |
| let I := {a : Fin d |
| let X (a : I) (z : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ)) := |
| fun u : ComponentIndex a.1 => F (z.1 (childComponentIndexVertex c a.1 u)) |
| (z.2 (childComponentIndexVertex c a.1 u)) |
| let Y (z : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ)) := |
| fun _ : Unit => F (z.1 r) (z.2 r) |
| let S (z : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ)) := z.1 r |
| have hF : Measurable (Function.uncurry F) := |
| ((PlantedInformation.measurable_spinSign.comp measurable_fst).const_mul _).add measurable_snd |
| have hS : Measurable S := (measurable_pi_apply r).comp measurable_fst |
| have hX (a : I) : Measurable (X a) := by |
| apply Measurable.of_eval |
| intro u |
| exact hF.comp (show Measurable (fun z : |
| (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) => |
| (z.1 (childComponentIndexVertex c a.1 u), z.2 (childComponentIndexVertex c a.1 u))) by fun_prop) |
| have hY : Measurable Y := by |
| apply Measurable.of_eval |
| intro u |
| exact hF.comp (show Measurable (fun z : |
| (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) => (z.1 r, z.2 r)) by fun_prop) |
| have hθ0 := tanh_nonneg_of_beta_nonneg hb |
| have hθ1 := Real.tanh_lt_one beta |
| have hθlo : -1 ≤ Real.tanh beta := by linarith |
| have hCI : iCondIndepFun (MeasurableSpace.comap S inferInstance) hS.comap_le X ρ := |
| hμ.iCondIndepFun_forward_child_observations hθlo hθ1.le (gaussianReal 0 t) hF c |
| have hOwn : CondIndepFun (MeasurableSpace.comap S inferInstance) hS.comap_le |
| Y (fun z a => X a z) ρ := |
| (hμ.condIndepFun_own_forward_child_observations hθlo hθ1.le |
| (gaussianReal 0 t) hF c).comp |
| (show Measurable (fun x : ℝ => fun _ : Unit => x) by fun_prop) measurable_id |
| have hfair (s : IsingSpin) : ρ {z | S z = s} = (1 / 2 : ENNReal) := |
| hμ.planted_spin_event_mass (fun _ => gaussianReal 0 t) r s |
| have hbound (a : I) : ∀ᵐ z ∂ρ, |
| |ρ[fun z => spinSign (S z) | MeasurableSpace.comap (X a) inferInstance] z| ≤ |
| Real.tanh beta := |
| hμ.selected_child_parent_abs_le hθ0 hθ1.le (gaussianReal 0 t) hF c a.1 |
| have hstrict (a : I) : ∀ᵐ z ∂ρ, |
| |ρ[fun z => spinSign (S z) | MeasurableSpace.comap (X a) inferInstance] z| < 1 := |
| (hbound a).mono (fun _ hz => hz.trans_lt hθ1) |
| have hsym (a : I) : (ρ.map (ρ[fun z => spinSign (S z) | |
| MeasurableSpace.comap (X a) inferInstance])).map (fun x : ℝ => -x) = |
| ρ.map (ρ[fun z => spinSign (S z) | MeasurableSpace.comap (X a) inferInstance]) := |
| hμ.selected_gaussian_posteriorLaw_neg_invariant t t (childComponentIndexVertex c a.1) r |
| have hOwnStrict : ∀ᵐ z ∂ρ, |
| |ρ[fun z => spinSign (S z) | MeasurableSpace.comap Y inferInstance] z| < 1 := by |
| filter_upwards [hμ.selected_singleton_gaussian_posterior hb t r] with z hz |
| change |selectedPosterior ρ F (fun _ : Unit => r) r z| < 1 |
| rw [hz] |
| exact Real.abs_tanh_lt_one _ |
| have hOwnSym : (ρ.map (ρ[fun z => spinSign (S z) | |
| MeasurableSpace.comap Y inferInstance])).map (fun x : ℝ => -x) = |
| ρ.map (ρ[fun z => spinSign (S z) | MeasurableSpace.comap Y inferInstance]) := |
| hμ.selected_gaussian_posteriorLaw_neg_invariant t t (fun _ : Unit => r) r |
| have hcard : Fintype.card I = d - 1 := by |
| dsimp [I] |
| simp [Fintype.card_subtype_compl] |
| obtain ⟨a, b, hab⟩ := Fintype.exists_pair_of_one_lt_card |
| (show 1 < Fintype.card I by rw [hcard]; omega) |
| have hfamily := condExp_own_and_finite_observation_secondMoment_le_sum_sub_loss |
| hS hX hCI hfair hstrict hsym hY hOwn hOwnStrict hOwnSym hab (hbound a) (hbound b) hθ1 |
| have hfield : MeasurableSpace.comap (fun z => (Y z, fun a => X a z)) inferInstance = |
| selectedObservationField F (componentIndexVertex (c := c)) := by |
| simpa only [Y, X, r, RootedBroadcast.branchRoot, selectedObservationField, |
| Prod.instMeasurableSpace, MeasurableSpace.comap_prodMk, |
| MeasurableSpace.comap_process_pi, iSup_const] using |
| (componentObservationField_eq_own_sup_children F c).symm |
| rw [hfield] at hfamily |
| have hchild (a : I) : (∫ z, (ρ[fun z => spinSign (S z) | |
| MeasurableSpace.comap (X a) inferInstance] z) ^ 2 ∂ρ) = |
| (Real.tanh beta) ^ 2 * gaussianComponentInformation μ t c := |
| hμ.selected_child_parent_secondMoment hθlo hθ1.le (gaussianReal 0 t) hF c a.1 |
| simp_rw [hchild] at hfamily |
| rw [Finset.sum_const, Finset.card_univ, hcard, nsmul_eq_mul] at hfamily |
| have hown := hμ.selected_singleton_information_le hb t r |
| change (∫ z, (ρ[fun z => spinSign (S z) | |
| MeasurableSpace.comap Y inferInstance] z) ^ 2 ∂ρ) ≤ (t : ℝ) + (t : ℝ) ^ 2 at hown |
| change gaussianComponentInformation μ t c ≤ _ at hfamily |
| nlinarith [hfamily, hown] |
|
|
| theorem HasFreeIsingLaw.gaussian_component_critical_information_upper |
| {d : ℕ} (hd : 3 ≤ d) {beta : ℝ} |
| {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (hcritical : ((d - 1 : ℕ) : ℝ) * (Real.tanh beta) ^ 2 = 1) |
| (t : ℝ≥0) (ht : (t : ℝ) ≤ 1) (c : Fin d) : |
| gaussianComponentInformation μ t c ≤ |
| Real.sqrt (1 / ((1 - (Real.tanh beta) ^ 2) * (Real.tanh beta) ^ 4)) * |
| Real.sqrt (t : ℝ) := by |
| have hθ0 := tanh_nonneg_of_beta_nonneg hb |
| have hθpos : 0 < Real.tanh beta := by |
| by_contra h |
| have hz : Real.tanh beta = 0 := le_antisymm (le_of_not_gt h) hθ0 |
| simp [hz] at hcritical |
| exact critical_information_upper_of_fusion hθpos (Real.tanh_lt_one beta) |
| hcritical t.coe_nonneg ht (hμ.gaussian_component_information_fusion hd hb t c) |
|
|
| end Problem367 |
|
|
| end |
|
|
| end OAI |
|
|
