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f61e547 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 | 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 // c ≠ a}, cavityMessage (Real.tanh beta)
(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 // c ≠ a}
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
end OAI
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