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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 | import OAI.Probability.FreeIsing.Estimates.ProgressivePrimitiveFubini
import OAI.Probability.FreeIsing.Estimates.SharedInnovationBounds
import OAI.Probability.FreeIsing.Posterior.FieldMoments
import OAI.Probability.FreeIsing.Posterior.ProgressiveIsingPosterior
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped NNReal
namespace Problem367
open PlantedSymmetry
theorem ae_observation_sub_bound_of_shared_progressive_innovation
{Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
{F G : Filtration ℝ≥0 mΩ} {m m' : ℝ≥0 → Ω → ℝ}
(hm : IsStronglyProgressive F m) (hm' : IsStronglyProgressive G m')
(hb : ∀ t, ∀ᵐ ω ∂P, |m t ω| ≤ 1)
(hb' : ∀ t, ∀ᵐ ω ∂P, |m' t ω| ≤ 1)
{x x' w : Ω → ℝ} (t : ℝ≥0)
(hx : ∀ᵐ ω ∂P, x ω = w ω + nnrealTimePrimitive m t ω)
(hx' : ∀ᵐ ω ∂P, x' ω = w ω + nnrealTimePrimitive m' t ω) :
∀ᵐ ω ∂P, |x ω - x' ω| ≤ 2 * (t : ℝ) := by
have hL := ae_lipschitzWith_nnrealTimePrimitive hm (C := 1) hb
have hL' := ae_lipschitzWith_nnrealTimePrimitive hm' (C := 1) hb'
filter_upwards [hL, hL', hx, hx'] with ω hω hω' hxω hxω'
have h1 : |nnrealTimePrimitive m t ω| ≤ (t : ℝ) := by
simpa only [Real.dist_eq, nnrealTimePrimitive, NNReal.coe_zero,
intervalIntegral.integral_same, sub_zero, NNReal.coe_one, one_mul,
NNReal.dist_eq, abs_of_nonneg t.coe_nonneg] using hω.dist_le_mul t 0
have h2 : |nnrealTimePrimitive m' t ω| ≤ (t : ℝ) := by
simpa only [Real.dist_eq, nnrealTimePrimitive, NNReal.coe_zero,
intervalIntegral.integral_same, sub_zero, NNReal.coe_one, one_mul,
NNReal.dist_eq, abs_of_nonneg t.coe_nonneg] using hω'.dist_le_mul t 0
rw [hxω, hxω']
have heq : (w ω + nnrealTimePrimitive m t ω) -
(w ω + nnrealTimePrimitive m' t ω) =
nnrealTimePrimitive m t ω - nnrealTimePrimitive m' t ω := by ring
rw [heq]
exact (abs_sub _ _).trans (by linarith)
theorem HasFreeIsingLaw.gaussianPosteriorField_copy_bound
{Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(t : ℝ≥0) (v : TreeVertex d)
{q q' : Ω → (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ)}
(hq : MeasurePreserving q P (μ.prod (gaussianNoiseLaw (TreeVertex d) t)))
(hq' : MeasurePreserving q' P (μ.prod (gaussianNoiseLaw (TreeVertex d) t)))
{F G : Filtration ℝ≥0 mΩ} {m m' : ℝ≥0 → Ω → ℝ}
(hm : IsStronglyProgressive F m) (hm' : IsStronglyProgressive G m')
(hb : ∀ s, ∀ᵐ ω ∂P, |m s ω| ≤ 1)
(hb' : ∀ s, ∀ᵐ ω ∂P, |m' s ω| ≤ 1)
{w : Ω → ℝ}
(hx : ∀ᵐ ω ∂P, observation t (q ω) v = w ω + nnrealTimePrimitive m t ω)
(hx' : ∀ᵐ ω ∂P, observation t (q' ω) v = w ω + nnrealTimePrimitive m' t ω) :
∀ᵐ ω ∂P, |gaussianPosteriorField μ t v (q ω) -
gaussianPosteriorField μ t v (q' ω)| ≤ 2 * (t : ℝ) + 2 * (d : ℝ) * beta := by
have hobs := ae_observation_sub_bound_of_shared_progressive_innovation
hm hm' hb hb' t hx hx'
have hfield := hμ.gaussian_posterior_field_bound hbeta t v
have hleft := hq.quasiMeasurePreserving.ae hfield
have hright := hq'.quasiMeasurePreserving.ae hfield
filter_upwards [hobs, hleft, hright] with ω hobsω hl hr
have hlb := hl.2
have hrb := hr.2
change |gaussianPosteriorField μ t v (q ω) - observation t (q ω) v| ≤
(d : ℝ) * beta at hlb
change |gaussianPosteriorField μ t v (q' ω) - observation t (q' ω) v| ≤
(d : ℝ) * beta at hrb
have htriangle := abs_sub_le (gaussianPosteriorField μ t v (q ω))
(observation t (q ω) v) (gaussianPosteriorField μ t v (q' ω))
have htriangle' := abs_sub_le (observation t (q ω) v)
(observation t (q' ω) v) (gaussianPosteriorField μ t v (q' ω))
rw [abs_sub_comm (observation t (q' ω) v)] at htriangle'
linarith
theorem HasFreeIsingLaw.gibbsField_bound_of_observation_identDistrib
{Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(t : ℝ≥0) (v : TreeVertex d) {X : Ω → TreeVertex d → ℝ}
(hX : IdentDistrib X (observation t) P
(μ.prod (gaussianNoiseLaw (TreeVertex d) t))) :
∀ᵐ ω ∂P, |Real.artanh (infiniteGibbsSpinMean beta v (X ω)) - X ω v| ≤
(d : ℝ) * beta := by
have ha : Measurable Real.artanh := by unfold Real.artanh; fun_prop
have hmeas := (ha.comp (measurable_infiniteGibbsSpinMean beta v)).sub
(measurable_pi_apply v)
apply hX.symm.ae_snd (measurableSet_le hmeas.abs measurable_const)
filter_upwards [hμ.infiniteGibbsSpinMean_eq_posterior t v,
hμ.gaussian_posterior_field_bound hbeta t v] with p heq hp
simp only [Pi.sub_apply, Function.comp_apply]
rw [heq]
exact hp.2
theorem HasFreeIsingLaw.gibbsField_copy_bound_of_shared_innovation
{Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
{F G : Filtration ℝ≥0 mΩ}
{X X' : TreeVertex d → ℝ≥0 → Ω → ℝ}
(hX : ∀ i, IsStronglyProgressive F (X i))
(hX' : ∀ i, IsStronglyProgressive G (X' i))
(t : ℝ≥0) (v : TreeVertex d)
(hlaw : IdentDistrib (fun ω i => X i t ω) (observation t) P
(μ.prod (gaussianNoiseLaw (TreeVertex d) t)))
(hlaw' : IdentDistrib (fun ω i => X' i t ω) (observation t) P
(μ.prod (gaussianNoiseLaw (TreeVertex d) t)))
{w : Ω → ℝ}
(hx : ∀ᵐ ω ∂P, X v t ω = w ω + nnrealTimePrimitive
(fun s p => infiniteGibbsSpinMean beta v (fun i => X i s p)) t ω)
(hx' : ∀ᵐ ω ∂P, X' v t ω = w ω + nnrealTimePrimitive
(fun s p => infiniteGibbsSpinMean beta v (fun i => X' i s p)) t ω) :
∀ᵐ ω ∂P,
|Real.artanh (infiniteGibbsSpinMean beta v (fun i => X i t ω)) -
Real.artanh (infiniteGibbsSpinMean beta v (fun i => X' i t ω))| ≤
2 * (t : ℝ) + 2 * (d : ℝ) * beta := by
have hm := isStronglyProgressive_infiniteGibbsSpinMean F beta v X hX
have hm' := isStronglyProgressive_infiniteGibbsSpinMean G beta v X' hX'
have hobs := ae_observation_sub_bound_of_shared_progressive_innovation hm hm'
(fun s => Filter.Eventually.of_forall
(fun p => infiniteGibbsSpinMean_abs_le_one beta v _))
(fun s => Filter.Eventually.of_forall
(fun p => infiniteGibbsSpinMean_abs_le_one beta v _)) t hx hx'
have hl := hμ.gibbsField_bound_of_observation_identDistrib hbeta t v hlaw
have hr := hμ.gibbsField_bound_of_observation_identDistrib hbeta t v hlaw'
filter_upwards [hobs, hl, hr] with ω hobsω hlω hrω
have htriangle := abs_sub_le
(Real.artanh (infiniteGibbsSpinMean beta v (fun i => X i t ω)))
(X v t ω)
(Real.artanh (infiniteGibbsSpinMean beta v (fun i => X' i t ω)))
have htriangle' := abs_sub_le (X v t ω) (X' v t ω)
(Real.artanh (infiniteGibbsSpinMean beta v (fun i => X' i t ω)))
rw [abs_sub_comm (X' v t ω)] at htriangle'
linarith
theorem HasFreeIsingLaw.gibbsField_copy_secondMoment_bounds
{Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
{F G : Filtration ℝ≥0 mΩ}
{X X' : TreeVertex d → ℝ≥0 → Ω → ℝ}
(hX : ∀ i, IsStronglyProgressive F (X i))
(hX' : ∀ i, IsStronglyProgressive G (X' i))
(t : ℝ≥0) (v : TreeVertex d)
(hlaw : IdentDistrib (fun ω i => X i t ω) (observation t) P
(μ.prod (gaussianNoiseLaw (TreeVertex d) t)))
(hlaw' : IdentDistrib (fun ω i => X' i t ω) (observation t) P
(μ.prod (gaussianNoiseLaw (TreeVertex d) t)))
{w : Ω → ℝ}
(hx : ∀ᵐ ω ∂P, X v t ω = w ω + nnrealTimePrimitive
(fun s p => infiniteGibbsSpinMean beta v (fun i => X i s p)) t ω)
(hx' : ∀ᵐ ω ∂P, X' v t ω = w ω + nnrealTimePrimitive
(fun s p => infiniteGibbsSpinMean beta v (fun i => X' i s p)) t ω) :
let U := fun ω =>
Real.artanh (infiniteGibbsSpinMean beta v (fun i => X i t ω)) -
Real.artanh (infiniteGibbsSpinMean beta v (fun i => X' i t ω))
MemLp U 2 P ∧ 0 ≤ (∫ ω, (U ω) ^ 2 ∂P) ∧
(∫ ω, (U ω) ^ 2 ∂P) ≤ (2 * (t : ℝ) + 2 * (d : ℝ) * beta) ^ 2 := by
dsimp only
have hb := hμ.gibbsField_copy_bound_of_shared_innovation hbeta hX hX'
t v hlaw hlaw' hx hx'
have ha : Measurable Real.artanh := by unfold Real.artanh; fun_prop
have hm := isStronglyProgressive_infiniteGibbsSpinMean F beta v X hX
have hm' := isStronglyProgressive_infiniteGibbsSpinMean G beta v X' hX'
have hmeas := (ha.comp (((hm.stronglyAdapted t).mono (F.le t)).measurable)).sub
(ha.comp (((hm'.stronglyAdapted t).mono (G.le t)).measurable))
exact discrepancy_secondMoment_bounds hmeas.aestronglyMeasurable hb
theorem progressive_memLp_two_on_horizon
{Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
{F : Filtration ℝ≥0 mΩ} {U : ℝ≥0 → Ω → ℝ}
(hU : IsStronglyProgressive F U) (T : ℝ≥0) (C : ℝ)
(hb : ∀ t ≤ T, ∀ᵐ ω ∂P, |U t ω| ≤ C) :
(∀ᵐ p ∂(volume.restrict (Ioc (0 : ℝ) (T : ℝ))).prod P,
|U p.1.toNNReal p.2| ≤ C) ∧
MemLp (fun p : ℝ × Ω => U p.1.toNNReal p.2) 2
((volume.restrict (Ioc (0 : ℝ) (T : ℝ))).prod P) := by
let ν : Measure ℝ := volume.restrict (Ioc 0 (T : ℝ))
have hclip : StronglyMeasurable (fun p : ℝ × Ω => U (min p.1.toNNReal T) p.2) :=
(stronglyMeasurable_clipped_nnreal_process hU T).mono
(sup_le_sup le_rfl (MeasurableSpace.comap_mono (F.le T)))
have hbc : ∀ᵐ p ∂ν.prod P, |U (min p.1.toNNReal T) p.2| ≤ C := by
apply (Measure.ae_prod_iff_ae_ae
(measurableSet_le hclip.measurable.abs measurable_const)).2
exact Filter.Eventually.of_forall fun r => hb _ (min_le_right r.toNNReal T)
have hc : MemLp (fun p : ℝ × Ω => U (min p.1.toNNReal T) p.2) 2 (ν.prod P) :=
MemLp.of_bound hclip.aestronglyMeasurable C
(by simpa only [Real.norm_eq_abs] using hbc)
have heq : (fun p : ℝ × Ω => U (min p.1.toNNReal T) p.2) =ᵐ[ν.prod P]
(fun p => U p.1.toNNReal p.2) := by
have hmem : ∀ᵐ p ∂ν.prod P, p.1 ∈ Ioc (0 : ℝ) (T : ℝ) :=
Measure.quasiMeasurePreserving_fst.ae (ae_restrict_mem measurableSet_Ioc)
filter_upwards [hmem] with p hp
have ht : p.1.toNNReal ≤ T := Real.toNNReal_le_iff_le_coe.mpr hp.2
simp only [min_eq_left ht]
refine ⟨?_, MemLp.ae_eq heq hc⟩
filter_upwards [heq, hbc] with p heq hp
rwa [← heq]
theorem HasFreeIsingLaw.gibbsField_copy_joint_bounds
{Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
{F : Filtration ℝ≥0 mΩ}
{X X' : TreeVertex d → ℝ≥0 → Ω → ℝ}
(hX : ∀ i, IsStronglyProgressive F (X i))
(hX' : ∀ i, IsStronglyProgressive F (X' i))
(T : ℝ≥0) (v : TreeVertex d)
(hlaw : ∀ t ≤ T, IdentDistrib (fun ω i => X i t ω) (observation t) P
(μ.prod (gaussianNoiseLaw (TreeVertex d) t)))
(hlaw' : ∀ t ≤ T, IdentDistrib (fun ω i => X' i t ω) (observation t) P
(μ.prod (gaussianNoiseLaw (TreeVertex d) t)))
{W : ℝ≥0 → Ω → ℝ}
(hx : ∀ t ≤ T, ∀ᵐ ω ∂P, X v t ω = W t ω + nnrealTimePrimitive
(fun s p => infiniteGibbsSpinMean beta v (fun i => X i s p)) t ω)
(hx' : ∀ t ≤ T, ∀ᵐ ω ∂P, X' v t ω = W t ω + nnrealTimePrimitive
(fun s p => infiniteGibbsSpinMean beta v (fun i => X' i s p)) t ω) :
let U := fun t ω =>
Real.artanh (infiniteGibbsSpinMean beta v (fun i => X i t ω)) -
Real.artanh (infiniteGibbsSpinMean beta v (fun i => X' i t ω))
(∀ᵐ p ∂(volume.restrict (Ioc (0 : ℝ) (T : ℝ))).prod P,
|U p.1.toNNReal p.2| ≤ 2 * (T : ℝ) + 2 * (d : ℝ) * beta) ∧
MemLp (fun p : ℝ × Ω => U p.1.toNNReal p.2) 2
((volume.restrict (Ioc (0 : ℝ) (T : ℝ))).prod P) := by
dsimp only
have ha : Measurable Real.artanh := by unfold Real.artanh; fun_prop
have hm := isStronglyProgressive_infiniteGibbsSpinMean F beta v X hX
have hm' := isStronglyProgressive_infiniteGibbsSpinMean F beta v X' hX'
have hU : IsStronglyProgressive F (fun t ω =>
Real.artanh (infiniteGibbsSpinMean beta v (fun i => X i t ω)) -
Real.artanh (infiniteGibbsSpinMean beta v (fun i => X' i t ω))) := by
intro t
exact ((ha.comp (hm t).measurable).sub
(ha.comp (hm' t).measurable)).stronglyMeasurable
apply progressive_memLp_two_on_horizon hU T (2 * (T : ℝ) + 2 * (d : ℝ) * beta)
intro t ht
filter_upwards [hμ.gibbsField_copy_bound_of_shared_innovation hbeta
hX hX' t v (hlaw t ht) (hlaw' t ht) (hx t ht) (hx' t ht)] with ω hω
exact hω.trans (by have htr : (t : ℝ) ≤ T := ht; linarith)
end Problem367
end
end OAI
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