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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 | 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 // 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 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
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