| import OAI.Probability.FreeIsing.Estimates.ComponentFusionRecurrence |
| import OAI.Probability.FreeIsing.Estimates.MessageFunction |
| import OAI.Probability.FreeIsing.Gaussian.FiniteSignalFieldRecursion |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory ProbabilityTheory |
| open scoped NNReal BigOperators |
|
|
| namespace Problem367 |
|
|
| theorem HasFreeIsingLaw.gaussian_component_field_recursion |
| {d : ℕ} (hd : 2 ≤ d) {beta : ℝ} |
| {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (t : ℝ≥0) (c : Fin d) : |
| let ρ := μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t) |
| let F : IsingSpin → ℝ → ℝ := fun s x => (t : ℝ) * spinSign s + x |
| ∀ᵐ z ∂ρ, |
| Real.artanh (selectedPosterior ρ F (componentIndexVertex (c := c)) |
| (RootedBroadcast.branchRoot c) z) = |
| PlantedSymmetry.observation t z (RootedBroadcast.branchRoot c) + |
| ∑ a : {a : Fin d |
| (Real.artanh (selectedPosterior ρ F (childComponentIndexVertex c a.1) |
| (prefixFlip c (RootedBroadcast.branchRoot a.1)) z)) := 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 hstrict (a : I) : ∀ᵐ z ∂ρ, |
| |ρ[fun z => spinSign (S z) | MeasurableSpace.comap (X a) inferInstance] z| < 1 := |
| (hμ.selected_child_parent_abs_le hθ0 hθ1.le (gaussianReal 0 t) hF c a.1).mono |
| (fun _ hz => hz.trans_lt hθ1) |
| have hOwnPost := hμ.selected_singleton_gaussian_posterior hb t r |
| have hOwnStrict : ∀ᵐ z ∂ρ, |
| |ρ[fun z => spinSign (S z) | MeasurableSpace.comap Y inferInstance] z| < 1 := by |
| filter_upwards [hOwnPost] with z hz |
| change |selectedPosterior ρ F (fun _ : Unit => r) r z| < 1 |
| rw [hz] |
| exact Real.abs_tanh_lt_one _ |
| let mAll := MeasurableSpace.comap (fun z a => X a z) inferInstance |
| let : MeasurableSpace ((TreeVertex d → IsingSpin) × (TreeVertex d → ℝ)) := |
| Prod.instMeasurableSpace |
| have hAll : Measurable (fun z a => X a z) := Measurable.of_eval hX |
| have hf : finiteSignalField (fun a => MeasurableSpace.comap (X a) inferInstance) |
| Finset.univ = mAll := by |
| simp [mAll, finiteSignalField, MeasurableSpace.comap_process_pi] |
| have hcard : Fintype.card I = d - 1 := by |
| dsimp [I] |
| simp [Fintype.card_subtype_compl] |
| have hne : (Finset.univ : Finset I).Nonempty := by |
| apply Finset.card_pos.mp |
| rw [Finset.card_univ, hcard] |
| omega |
| have hFields := iIndep_observationFields_cond_spin hS hX hCI hfair |
| have hAllStrict : ∀ᵐ z ∂ρ, |
| |ρ[fun z => spinSign (S z) | mAll] z| < 1 := by |
| rw [← hf] |
| exact abs_condExp_spin_finiteSignalField_lt_one (fun a => (hX a).comap_le) |
| hS hstrict hfair hFields Finset.univ hne |
| have hPairInd (s : IsingSpin) : |
| Indep (MeasurableSpace.comap Y inferInstance) mAll (cond ρ {z | S z = s}) := |
| indepFun_cond_fiber_of_condIndepFun hS hY hAll hOwn s |
| have hfield : MeasurableSpace.comap Y inferInstance ⊔ mAll = |
| selectedObservationField F (componentIndexVertex (c := c)) := by |
| simpa only [Y, X, r, RootedBroadcast.branchRoot, mAll, selectedObservationField, |
| MeasurableSpace.comap_process_pi, iSup_const] using |
| (componentObservationField_eq_own_sup_children F c).symm |
| have hFullStrict : ∀ᵐ z ∂ρ, |
| |selectedPosterior ρ F (componentIndexVertex (c := c)) r z| < 1 := by |
| have h := abs_condExp_spin_sup_lt_one hY.comap_le hAll.comap_le hS |
| hOwnStrict hAllStrict hfair hPairInd |
| rw [hfield] at h |
| exact h |
| have hChildStrict (a : I) : ∀ᵐ z ∂ρ, |
| |selectedPosterior ρ F (childComponentIndexVertex c a.1) |
| (prefixFlip c (RootedBroadcast.branchRoot a.1)) z| < 1 := by |
| exact (childComponentPosterior_identDistrib hμ (gaussianReal 0 t) hF c a.1).symm.ae_snd |
| (measurableSet_lt measurable_abs measurable_const) hFullStrict |
| have hChild (a : I) : ∀ᵐ z ∂ρ, |
| Real.artanh (ρ[fun z => spinSign (S z) | |
| MeasurableSpace.comap (X a) inferInstance] z) = |
| cavityMessage (Real.tanh beta) (Real.artanh (selectedPosterior ρ F |
| (childComponentIndexVertex c a.1) (prefixFlip c (RootedBroadcast.branchRoot a.1)) z)) := by |
| filter_upwards [hμ.selectedPosterior_parent_childComponent hθlo hθ1.le |
| (gaussianReal 0 t) hF c a.1, hChildStrict a] with z hz hi |
| change selectedPosterior ρ F (childComponentIndexVertex c a.1) r z = |
| Real.tanh beta * selectedPosterior ρ F (childComponentIndexVertex c a.1) |
| (prefixFlip c (RootedBroadcast.branchRoot a.1)) z at hz |
| change Real.artanh (selectedPosterior ρ F (childComponentIndexVertex c a.1) r z) = _ |
| rw [hz, cavityMessage, Real.tanh_artanh (abs_lt.mp hi)] |
| have hSum := artanh_condExp_spin_finiteSignalField (fun a => (hX a).comap_le) |
| hS hstrict hfair hFields Finset.univ hne |
| rw [hf] at hSum |
| have hPair := artanh_condExp_spin_sup hY.comap_le hAll.comap_le hS |
| hOwnStrict hAllStrict hfair hPairInd |
| rw [hfield] at hPair |
| filter_upwards [hPair, hSum, hOwnPost, ae_all_iff.2 hChild] with z hp hs ho hc |
| change Real.artanh (selectedPosterior ρ F (componentIndexVertex (c := c)) r z) = _ at hp ⊢ |
| rw [hp] |
| change Real.artanh (selectedPosterior ρ F (fun _ : Unit => r) r z) + _ = _ |
| rw [ho, Real.artanh_tanh, hs] |
| congr 1 |
| exact Finset.sum_congr rfl (fun a _ => hc a) |
|
|
| end Problem367 |
|
|
| end |
|
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| end OAI |
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|
