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b317718 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 | import OAI.Probability.FreeIsing.Trees.ConeSelectedFusion
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367
open PathSideBranches
theorem condExp_regression_of_le
{Ω : Type*} [m : MeasurableSpace Ω] (P : Measure Ω) [IsFiniteMeasure P]
{m₁ m₂ : MeasurableSpace Ω} (h₁₂ : m₁ ≤ m₂) (h₂ : m₂ ≤ m)
{f g : Ω → ℝ} {theta : ℝ}
(h : P[f | m₂] =ᵐ[P] fun z => theta * (P[g | m₂]) z) :
P[f | m₁] =ᵐ[P] fun z => theta * (P[g | m₁]) z := by
have hreg := condExp_congr_ae (m := m₁) h
have hmul := condExp_smul (μ := P) theta (P[g | m₂]) m₁
filter_upwards [hreg, hmul, condExp_condExp_of_le (f := f) h₁₂ h₂,
condExp_condExp_of_le (f := g) h₁₂ h₂] with z hz hm hf hg
simpa only [Pi.smul_apply, smul_eq_mul, hf, hg] using hz.trans hm
theorem HasFreeIsingLaw.selectedPosterior_parent_subset_cone
{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))
(r : TreeVertex d) (hr : r.1 ≠ [])
(T : Set (TreeVertex d)) (hT : T ⊆ cone r) :
let P := μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν))
selectedPosterior P F (Subtype.val : T → TreeVertex d) (parentVertex r) =ᵐ[P]
fun z => theta * selectedPosterior P F (Subtype.val : T → TreeVertex d) r z := by
let := hμ.isProbabilityMeasure
have hle : selectedObservationField F (Subtype.val : T → TreeVertex d) ≤
selectedObservationField F (Subtype.val : cone r → TreeVertex d) := by
apply selectedObservationField_le_of_range_subset F
rintro u ⟨v, rfl⟩
exact ⟨⟨v.1, hT v.2⟩, rfl⟩
exact condExp_regression_of_le _ hle
(selectedObservationField_le hF (Subtype.val : cone r → TreeVertex d))
(hμ.selectedPosterior_parent_cone h0 h1 ν hF r hr)
theorem HasFreeIsingLaw.selected_gaussian_outside_subset_cone_fusion
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(t : ℝ≥0) (r : TreeVertex d) (hr : r.1 ≠ [])
(S T : Set (TreeVertex d)) (hS : S ⊆ (cone r)ᶜ) (hT : T ⊆ cone r)
(hp : parentVertex r ∈ S) (hq : r ∈ T) :
let P := μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t)
let F : IsingSpin → ℝ → ℝ := fun s z => (t : ℝ) * spinSign s + z
selectedPosterior P F (Subtype.val : (S ∪ T : Set (TreeVertex d)) → TreeVertex d) (parentVertex r) =ᵐ[P]
fun z => binaryFusion
(selectedPosterior P F (Subtype.val : S → TreeVertex d) (parentVertex r) z)
(Real.tanh beta * selectedPosterior P F
(Subtype.val : T → TreeVertex d) r z) := by
let := hμ.isProbabilityMeasure
let P := μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t)
let F : IsingSpin → ℝ → ℝ := fun s z => (t : ℝ) * spinSign s + z
let X := fun z : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) =>
fun i : S => F (z.1 i.1) (z.2 i.1)
let Y := fun z : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) =>
fun i : T => F (z.1 i.1) (z.2 i.1)
have hF : Measurable (Function.uncurry F) :=
((PlantedInformation.measurable_spinSign.comp measurable_fst).const_mul
(t : ℝ)).add measurable_snd
have hX : Measurable X := by
apply Measurable.of_eval
intro i
exact hF.comp (show Measurable (fun z :
(TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) =>
(z.1 i.1, z.2 i.1)) from by fun_prop)
have hY : Measurable Y := by
apply Measurable.of_eval
intro i
exact hF.comp (show Measurable (fun z :
(TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) =>
(z.1 i.1, z.2 i.1)) from by fun_prop)
have hXout : Measurable[noisyRegionSigma (A := ℝ) (cone r)ᶜ] X :=
(comap_measurable X).mono
((selectedObservationField_le_noisyRegionSigma hF S).trans
(noisyRegionSigma_mono hS)) le_rfl
have hYin : Measurable[noisyRegionSigma (A := ℝ) (cone r)] Y :=
(comap_measurable Y).mono
((selectedObservationField_le_noisyRegionSigma hF T).trans
(noisyRegionSigma_mono hT)) le_rfl
have hInd := hμ.condIndepFun_noisy_cone_observables
(Real.neg_one_lt_tanh beta).le (Real.tanh_lt_one beta).le
(gaussianReal 0 t) r hr X Y hXout hYin
have hreg : selectedPosterior P F (Subtype.val : T → TreeVertex d)
(parentVertex r) =ᵐ[P] fun z => Real.tanh beta *
selectedPosterior P F (Subtype.val : T → TreeVertex d) r z :=
hμ.selectedPosterior_parent_subset_cone
(Real.neg_one_lt_tanh beta).le (Real.tanh_lt_one beta).le
(gaussianReal 0 t) hF r hr T hT
have hL : ∀ᵐ z ∂P,
|selectedPosterior P F (Subtype.val : S → TreeVertex d) (parentVertex r) z| < 1 := by
filter_upwards [hμ.selected_gaussian_posterior_field_bound hbeta t
(Subtype.val : S → TreeVertex d) (parentVertex r) ⟨⟨_, hp⟩, rfl⟩] with z hz
exact abs_lt.mpr hz.1
have hQ : ∀ᵐ z ∂P,
|selectedPosterior P F (Subtype.val : T → TreeVertex d) r z| < 1 := by
filter_upwards [hμ.selected_gaussian_posterior_field_bound hbeta t
(Subtype.val : T → TreeVertex d) r ⟨⟨r, hq⟩, rfl⟩] with z hz
exact abs_lt.mpr hz.1
have hR : ∀ᵐ z ∂P,
|selectedPosterior P F (Subtype.val : T → TreeVertex d) (parentVertex r) z| < 1 := by
filter_upwards [hreg, hQ] with z hz hq
rw [hz, abs_mul]
exact (mul_le_mul_of_nonneg_left hq.le (abs_nonneg (Real.tanh beta))).trans_lt
(by simpa using Real.abs_tanh_lt_one beta)
have h := condExp_spin_comap_sup_eq_binaryFusion
((measurable_pi_apply (parentVertex r)).comp measurable_fst) hX hY
(hμ.planted_spin_event_mass (fun _ => gaussianReal 0 t) _) hInd hL hR
dsimp only
unfold selectedPosterior
rw [selectedObservationField_region_union]
filter_upwards [h, hreg] with z hz hregz
exact hz.trans (congrArg (binaryFusion
(selectedPosterior P F (Subtype.val : S → TreeVertex d) (parentVertex r) z)) hregz)
theorem HasFreeIsingLaw.selected_gaussian_outside_subset_cone_field
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(t : ℝ≥0) (r : TreeVertex d) (hr : r.1 ≠ [])
(S T : Set (TreeVertex d)) (hS : S ⊆ (cone r)ᶜ) (hT : T ⊆ cone r)
(hp : parentVertex r ∈ S) (hq : r ∈ T) :
let P := μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t)
let F : IsingSpin → ℝ → ℝ := fun s z => (t : ℝ) * spinSign s + z
(fun z => Real.artanh (selectedPosterior P F
(Subtype.val : (S ∪ T : Set (TreeVertex d)) → TreeVertex d) (parentVertex r) z)) =ᵐ[P]
fun z => Real.artanh (selectedPosterior P F
(Subtype.val : S → TreeVertex d) (parentVertex r) z) +
statisticMessage (Real.tanh beta) (selectedPosterior P F
(Subtype.val : T → TreeVertex d) r z) := by
have h := hμ.selected_gaussian_outside_subset_cone_fusion hbeta t r hr S T hS hT hp hq
have hL := hμ.selected_gaussian_posterior_field_bound hbeta t
(Subtype.val : S → TreeVertex d) (parentVertex r) ⟨⟨_, hp⟩, rfl⟩
have hQ := hμ.selected_gaussian_posterior_field_bound hbeta t
(Subtype.val : T → TreeVertex d) r ⟨⟨r, hq⟩, rfl⟩
filter_upwards [h, hL, hQ] with z hz hl hq
rw [hz]
exact artanh_binaryFusion (abs_lt.mpr hl.1)
(binaryFusion_abs_mul_lt_one (Real.abs_tanh_lt_one beta) (abs_lt.mpr hq.1))
theorem HasFreeIsingLaw.selected_gaussian_subset_cone_field_difference
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(t : ℝ≥0) (r : TreeVertex d) (hr : r.1 ≠ [])
(S T U : Set (TreeVertex d)) (hS : S ⊆ (cone r)ᶜ)
(hT : T ⊆ cone r) (hU : U ⊆ cone r)
(hp : parentVertex r ∈ S) (ht : r ∈ T) (hu : r ∈ U) :
let P := μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t)
let F : IsingSpin → ℝ → ℝ := fun s z => (t : ℝ) * spinSign s + z
let field := fun (V : Set (TreeVertex d)) (u : TreeVertex d) z =>
Real.artanh (selectedPosterior P F (Subtype.val : V → TreeVertex d) u z)
∀ᵐ z ∂P,
|field (S ∪ T) (parentVertex r) z - field (S ∪ U) (parentVertex r) z| ≤
Real.tanh beta * |field T r z - field U r z| ∧
|field (S ∪ T) (parentVertex r) z - field (S ∪ U) (parentVertex r) z| ≤
2 * beta := by
have hFT := hμ.selected_gaussian_outside_subset_cone_field hbeta t r hr S T hS hT hp ht
have hFU := hμ.selected_gaussian_outside_subset_cone_field hbeta t r hr S U hS hU hp hu
have hQT := hμ.selected_gaussian_posterior_field_bound hbeta t
(Subtype.val : T → TreeVertex d) r ⟨⟨r, ht⟩, rfl⟩
have hQU := hμ.selected_gaussian_posterior_field_bound hbeta t
(Subtype.val : U → TreeVertex d) r ⟨⟨r, hu⟩, rfl⟩
have hθ0 : 0 ≤ Real.tanh beta := by
rw [Real.tanh_eq_sinh_div_cosh]
exact div_nonneg (Real.sinh_nonneg_iff.mpr hbeta) (Real.cosh_pos beta).le
filter_upwards [hFT, hFU, hQT, hQU] with z hft hfu hqt hqu
rw [hft, hfu, add_sub_add_left_eq_sub]
constructor
· have h := (lipschitzWith_cavityMessage hθ0 (Real.tanh_lt_one beta).le).dist_le_mul
(Real.artanh (selectedPosterior
(μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t))
(fun s z => (t : ℝ) * spinSign s + z) (Subtype.val : T → TreeVertex d) r z))
(Real.artanh (selectedPosterior
(μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t))
(fun s z => (t : ℝ) * spinSign s + z) (Subtype.val : U → TreeVertex d) r z))
convert h using 1 <;>
simp only [Real.dist_eq, cavityMessage, Real.tanh_artanh hqt.1,
Real.tanh_artanh hqu.1, statisticMessage]
rfl
· have ht' := statisticMessage_abs_le hθ0 (Real.tanh_lt_one beta)
(abs_lt.mpr hqt.1).le
have hu' := statisticMessage_abs_le hθ0 (Real.tanh_lt_one beta)
(abs_lt.mpr hqu.1).le
rw [Real.artanh_tanh] at ht' hu'
exact (abs_sub _ _).trans (by linarith)
end Problem367
end
end OAI
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