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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 | import OAI.Probability.FreeIsing.Energy.DiscrepancyRowEnergy
import OAI.Probability.FreeIsing.Energy.NormalizedInfiniteTimeEnergy
import OAI.Probability.FreeIsing.Ising.ObservationComparison
import OAI.Probability.FreeIsing.Ising.ObservationPosteriorTransport
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped NNReal ENNReal Topology
namespace Problem367.IsingCopyEnergy
open IsingObservationInnovation IsingObservationFactor
def responseRow {d : ℕ} (beta : ℝ) (v w : TreeVertex d) (p : ℝ × Paths d) : ℝ :=
infiniteGibbsResponse beta v w (fun i => p.2 i p.1.toNNReal)
theorem measurable_responseRow {d : ℕ} (beta : ℝ) (v w : TreeVertex d) :
Measurable (responseRow beta v w) := by
apply (measurable_infiniteGibbsResponse beta v w).comp
apply Measurable.of_eval
intro i
exact continuous_eval.measurable.comp
(((measurable_pi_apply i).comp measurable_snd).prodMk
(measurable_real_toNNReal.comp measurable_fst))
theorem observation_extended_energy_le
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(v : TreeVertex d) {T : ℝ} (hT : 0 ≤ T) :
(∫⁻ p, ∑' w, ENNReal.ofReal (responseRow beta v w p ^ 2)
∂(volume.restrict (Ioc 0 T)).prod (observationLaw μ)) ≤
ENNReal.ofReal (2 * ((d : ℝ) * beta + T + Real.sqrt T)) := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
let : IsProbabilityMeasure (labelMeasure d) := by unfold labelMeasure; infer_instance
let Q := (volume.restrict (Ioc (0 : ℝ) T)).prod (μ.prod (labelMeasure d))
let R := (volume.restrict (Ioc (0 : ℝ) T)).prod (observationLaw μ)
let f : ℝ × Paths d → ℝ≥0∞ := fun p => ∑' w, ENNReal.ofReal (responseRow beta v w p ^ 2)
have hf : Measurable f := Measurable.tsum (fun w =>
((measurable_responseRow beta v w).pow_const 2).ennreal_ofReal)
have hmp : MeasurePreserving (Prod.map id uniformObservation) Q R :=
(MeasurePreserving.id _).prod (measurePreserving_uniformObservation μ)
have henergy := NormalizedInfiniteTimeEnergy.lintegral_infiniteRow_le hμ hb
(fun _ : TreeVertex d => uniform01)
(fun _ => WienerSampler.sample_isBrownianReal.toIsPreBrownianReal)
(fun _ t => (continuous_eval_const t).measurable.comp WienerSampler.measurable_sample)
(fun _ => WienerSampler.measurable_sample_eval) v hT
have heq : (fun p => f (Prod.map id uniformObservation p)) =ᵐ[Q]
(fun p => ∑' w, ENNReal.ofReal
(NormalizedInfiniteTimeEnergy.infiniteRow beta v
(fun _ t u => WienerSampler.sample u t) w p ^ 2)) := by
have hmem : ∀ᵐ p ∂Q, p.1 ∈ Ioc (0 : ℝ) T :=
Measure.quasiMeasurePreserving_fst.ae (ae_restrict_mem measurableSet_Ioc)
filter_upwards [hmem] with p hp
apply tsum_congr
intro w
apply congrArg ENNReal.ofReal
apply congrArg (fun r : ℝ => r ^ 2)
change infiniteGibbsResponse beta v w (fun i => uniformObservation p.2 i p.1.toNNReal) = _
apply congrArg (infiniteGibbsResponse beta v w)
funext i
change WienerSampler.sample (p.2.2 i) p.1.toNNReal +
(p.1.toNNReal : ℝ) * spinSign (p.2.1 i) =
p.1 * spinSign (p.2.1 i) + WienerSampler.sample (p.2.2 i) p.1.toNNReal
rw [Real.coe_toNNReal p.1 hp.1.le, add_comm]
calc
_ = ∫⁻ p, f (Prod.map id uniformObservation p) ∂Q := (hmp.lintegral_comp hf).symm
_ = _ := lintegral_congr_ae heq
_ ≤ _ := henergy
theorem observation_row_energy_integrable
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(v : TreeVertex d) {T : ℝ} (hT : 0 ≤ T) :
(∀ᵐ p ∂(volume.restrict (Ioc 0 T)).prod (observationLaw μ),
Summable (fun w => responseRow beta v w p ^ 2)) ∧
Integrable (fun p => ∑' w, responseRow beta v w p ^ 2)
((volume.restrict (Ioc 0 T)).prod (observationLaw μ)) ∧
(∫ p, ∑' w, responseRow beta v w p ^ 2
∂(volume.restrict (Ioc 0 T)).prod (observationLaw μ)) ≤
2 * ((d : ℝ) * beta + T + Real.sqrt T) := by
let P := (volume.restrict (Ioc (0 : ℝ) T)).prod (observationLaw μ)
have he := observation_extended_energy_le hμ hb v hT
rw [lintegral_tsum (fun w =>
((measurable_responseRow beta v w).pow_const 2).ennreal_ofReal.aemeasurable)] at he
have hm (w : TreeVertex d) : AEStronglyMeasurable (responseRow beta v w) P :=
(measurable_responseRow beta v w).aestronglyMeasurable
exact ⟨ae_summable_sq_of_tsum_lintegral_sq_ne_top P hm
(ne_top_of_le_ne_top ENNReal.ofReal_ne_top he),
integral_tsum_sq_le_of_tsum_lintegral_sq_le P hm (by positivity) he⟩
abbrev Joint (d : ℕ) := Paths d × (Paths d × Paths d)
def projection {d : ℕ} (j : Bool) (q : Joint d) : Paths d :=
if j then q.2.1 else q.2.2
private theorem projection_measurePreserving
{d : ℕ} (beta : ℝ) (μ : Measure (TreeVertex d → IsingSpin))
[IsProbabilityMeasure μ] (j : Bool) :
MeasurePreserving (projection (d := d) j) (copies beta μ) (observationLaw μ) := by
obtain ⟨hl, hr⟩ := IsingConditionalCopyBounds.measurePreserving_observation_copies
(beta := beta) (μ := μ)
(rho := (observationLaw μ).map (innovationMap beta)) (eta := kernel beta μ) kernel_graphLaw
cases j
· exact hr
· exact hl
def timeProjection {d : ℕ} (j : Bool) (p : ℝ × Joint d) : ℝ × Paths d :=
(p.1, projection j p.2)
private theorem timeProjection_measurePreserving
{d : ℕ} (beta : ℝ) (μ : Measure (TreeVertex d → IsingSpin))
[IsProbabilityMeasure μ] (j : Bool) (T : ℝ) :
MeasurePreserving (timeProjection (d := d) j)
((volume.restrict (Ioc (0 : ℝ) T)).prod (copies beta μ))
((volume.restrict (Ioc (0 : ℝ) T)).prod (observationLaw μ)) :=
(MeasurePreserving.id _).prod (projection_measurePreserving beta μ j)
theorem copy_row_energy_integrable
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
[IsProbabilityMeasure μ]
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(v : TreeVertex d) (j : Bool) {T : ℝ} (hT : 0 ≤ T) :
(∀ᵐ p ∂(volume.restrict (Ioc 0 T)).prod (copies beta μ),
Summable (fun w => responseRow beta v w (timeProjection j p) ^ 2)) ∧
Integrable (fun p => ∑' w, responseRow beta v w (timeProjection j p) ^ 2)
((volume.restrict (Ioc 0 T)).prod (copies beta μ)) ∧
(∫ p, ∑' w, responseRow beta v w (timeProjection j p) ^ 2
∂(volume.restrict (Ioc 0 T)).prod (copies beta μ)) ≤
2 * ((d : ℝ) * beta + T + Real.sqrt T) := by
have ho := observation_row_energy_integrable hμ hb v hT
have hp := timeProjection_measurePreserving beta μ j T
refine ⟨hp.quasiMeasurePreserving.ae ho.1, hp.integrable_comp_of_integrable ho.2.1, ?_⟩
simpa only [Function.comp_def] using
(hp.hasLaw.integral_comp ho.2.1.aestronglyMeasurable).le.trans ho.2.2
def density {d : ℕ} (beta : ℝ) (v : TreeVertex d) (t : ℝ) (q : Joint d) : ℝ :=
DiscrepancyEnergy.density
(fun w q => responseRow beta v w (t, q.2.1))
(fun w q => responseRow beta v w (t, q.2.2))
(fun q => field beta v t.toNNReal q.2.1 - field beta v t.toNNReal q.2.2)
(fun q => posterior beta v t.toNNReal q.2.1)
(fun q => posterior beta v t.toNNReal q.2.2) q
theorem density_integrable
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
[IsProbabilityMeasure μ]
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(v : TreeVertex d) (T : ℝ≥0) :
Integrable (fun p : ℝ × Joint d => density beta v p.1 p.2)
((volume.restrict (Ioc (0 : ℝ) T)).prod (copies beta μ)) := by
have hl := copy_row_energy_integrable hμ hb v true T.coe_nonneg
have hr := copy_row_energy_integrable hμ hb v false T.coe_nonneg
have hmap (j : Bool) : Measurable (fun p : ℝ × Joint d =>
(p.1.toNNReal, projection j p.2)) :=
(measurable_real_toNNReal.comp measurable_fst).prodMk
((projection_measurePreserving beta μ j).measurable.comp measurable_snd)
have hbound := (IsingConditionalCopyBounds.fieldDifference_joint_bounds
hμ hb kernel_graphLaw T v).1
apply DiscrepancyEnergy.integrable_noise_add_drift
(c := fun w p => responseRow beta v w (timeProjection true p))
(c' := fun w p => responseRow beta v w (timeProjection false p))
(H := 2 * (T : ℝ) + 2 * (d : ℝ) * beta)
(fun w => ((measurable_responseRow beta v w).comp
(timeProjection_measurePreserving beta μ true T).measurable).aestronglyMeasurable)
(fun w => ((measurable_responseRow beta v w).comp
(timeProjection_measurePreserving beta μ false T).measurable).aestronglyMeasurable)
(((measurable_field_joint beta v).comp (hmap true)).sub
((measurable_field_joint beta v).comp (hmap false))).aestronglyMeasurable
(((measurable_posterior_joint beta v).comp (hmap true)).aestronglyMeasurable)
(((measurable_posterior_joint beta v).comp (hmap false)).aestronglyMeasurable)
hl.2.1 hr.2.1 (hl.1.and hr.1)
filter_upwards [hbound] with p hp
exact ⟨hp, posterior_norm_le_one beta v p.1.toNNReal p.2.2.1,
posterior_norm_le_one beta v p.1.toNNReal p.2.2.2⟩
theorem ae_integral_density_eq
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
[IsProbabilityMeasure μ]
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(v : TreeVertex d) (T : ℝ≥0) :
∀ᵐ t ∂volume.restrict (Icc (0 : ℝ) T),
(∫ q, density beta v t q ∂copies beta μ) =
(∑' w, ∫ q, (responseRow beta v w (t, q.2.1) -
responseRow beta v w (t, q.2.2)) ^ 2 ∂copies beta μ) +
2 * ∫ q, (field beta v t.toNNReal q.2.1 - field beta v t.toNNReal q.2.2) *
(posterior beta v t.toNNReal q.2.1 * (∑' w, responseRow beta v w (t, q.2.1) ^ 2) -
posterior beta v t.toNNReal q.2.2 * (∑' w, responseRow beta v w (t, q.2.2) ^ 2))
∂copies beta μ := by
have hl := copy_row_energy_integrable hμ hb v true T.coe_nonneg
have hr := copy_row_energy_integrable hμ hb v false T.coe_nonneg
rw [← restrict_Ioc_eq_restrict_Icc]
filter_upwards [hl.2.1.prod_right_ae, hr.2.1.prod_right_ae,
Measure.ae_ae_of_ae_prod hl.1, Measure.ae_ae_of_ae_prod hr.1] with t hAl hAr hsl hsr
have hmap (j : Bool) : Measurable (fun q : Joint d => (t.toNNReal, projection j q)) :=
measurable_const.prodMk (projection_measurePreserving beta μ j).measurable
have hrow (j : Bool) (w : TreeVertex d) :
Measurable (fun q : Joint d => responseRow beta v w (t, projection j q)) :=
(measurable_responseRow beta v w).comp
(measurable_const.prodMk (projection_measurePreserving beta μ j).measurable)
apply DiscrepancyEnergy.integral_noise_add_drift
(H := 2 * (t.toNNReal : ℝ) + 2 * (d : ℝ) * beta)
(fun w => (hrow true w).aestronglyMeasurable)
(fun w => (hrow false w).aestronglyMeasurable)
(((measurable_field_joint beta v).comp (hmap true)).sub
((measurable_field_joint beta v).comp (hmap false))).aestronglyMeasurable
(((measurable_posterior_joint beta v).comp (hmap true)).aestronglyMeasurable)
(((measurable_posterior_joint beta v).comp (hmap false)).aestronglyMeasurable)
hAl hAr (hsl.and hsr)
filter_upwards [IsingConditionalCopyBounds.ae_fieldDifference_bound
hμ hb kernel_graphLaw t.toNNReal v] with q hq
exact ⟨hq, posterior_norm_le_one beta v t.toNNReal q.2.1,
posterior_norm_le_one beta v t.toNNReal q.2.2⟩
end Problem367.IsingCopyEnergy
end
end OAI
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