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import OAI.Probability.FreeIsing.Ising.ObservationComparison
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped NNReal ENNReal Topology
namespace Problem367.NormalizedObservationRows
open IsingObservationInnovation IsingObservationFactor GaussianTreeResponse
variable {d : ℕ}
def rowError (beta : ℝ) (D : Finset (TreeVertex d)) (v w : TreeVertex d)
(p : ℝ × Paths d) : ℝ :=
(if w ∈ D then response beta D v w (fun i => p.2 i p.1.toNNReal) else 0) -
infiniteGibbsResponse beta v w (fun i => p.2 i p.1.toNNReal)
theorem measurable_rowError (beta : ℝ) (D : Finset (TreeVertex d))
(v w : TreeVertex d) : Measurable (rowError beta D v w) := by
classical
have hm : Measurable (fun p : ℝ × Paths d => fun i => p.2 i p.1.toNNReal) :=
Measurable.of_eval fun i => continuous_eval.measurable.comp
(((measurable_pi_apply i).comp measurable_snd).prodMk
(measurable_real_toNNReal.comp measurable_fst))
unfold rowError
by_cases hw : w ∈ D
· simp only [ite_eq_left hw]
exact ((measurable_response beta D v w).comp hm).sub
((measurable_infiniteGibbsResponse beta v w).comp hm)
· simp only [ite_eq_right hw]
exact measurable_const.sub ((measurable_infiniteGibbsResponse beta v w).comp hm)
theorem rowError_uniformObservation (beta : ℝ) (D : Finset (TreeVertex d))
(v w : TreeVertex d) (p : ℝ × IsingInnovation.State d ℝ) (ht : 0 ≤ p.1) :
rowError beta D v w (p.1, uniformObservation p.2) =
NormalizedInfiniteTimeEnergy.finiteRow beta D v
(fun _ t u => WienerSampler.sample u t) w p -
NormalizedInfiniteTimeEnergy.infiniteRow beta v
(fun _ t u => WienerSampler.sample u t) w p := by
have heq : (fun i => uniformObservation p.2 i p.1.toNNReal) =
(fun i => p.1 * spinSign (p.2.1 i) + WienerSampler.sample (p.2.2 i) p.1.toNNReal) := by
funext i
change WienerSampler.sample (p.2.2 i) p.1.toNNReal +
(p.1.toNNReal : ℝ) * spinSign (p.2.1 i) = _
rw [Real.coe_toNNReal p.1 ht, add_comm]
unfold rowError NormalizedInfiniteTimeEnergy.finiteRow
NormalizedInfiniteTimeEnergy.infiniteRow GaussianTreeResponse.observedRow
rw [heq]
rfl
theorem rowError_lintegral_eq
(beta : ℝ) (μ : Measure (TreeVertex d → IsingSpin)) [IsProbabilityMeasure μ]
(D : Finset (TreeVertex d)) (v : TreeVertex d) (T : ℝ) :
(∫⁻ p, ∑' w, ENNReal.ofReal (rowError beta D v w p ^ 2)
∂(volume.restrict (Ioc 0 T)).prod (observationLaw μ)) =
(∫⁻ p, ∑' w, ENNReal.ofReal
((NormalizedInfiniteTimeEnergy.finiteRow beta D v
(fun _ t u => WienerSampler.sample u t) w p -
NormalizedInfiniteTimeEnergy.infiniteRow beta v
(fun _ t u => WienerSampler.sample u t) w p) ^ 2)
∂(volume.restrict (Ioc 0 T)).prod (μ.prod (labelMeasure d))) := by
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 (rowError beta D v w p ^ 2)
have hf : Measurable f := Measurable.tsum fun w =>
((measurable_rowError beta D v w).pow_const 2).ennreal_ofReal
have hobs : MeasurePreserving uniformObservation (μ.prod (labelMeasure d))
(observationLaw μ) := ⟨measurable_uniformObservation, rfl⟩
have hp : MeasurePreserving (Prod.map id uniformObservation) Q R :=
(MeasurePreserving.id _).prod hobs
change (∫⁻ p, f p ∂ R) = _
rw [← hp.lintegral_comp hf]
apply lintegral_congr_ae
have ht : ∀ᵐ p ∂Q, p.1 ∈ Ioc (0 : ℝ) T :=
Measure.quasiMeasurePreserving_fst.ae (ae_restrict_mem measurableSet_Ioc)
filter_upwards [ht] with p hp
apply tsum_congr
intro w
exact congrArg (fun r : ℝ => ENNReal.ofReal (r ^ 2))
(rowError_uniformObservation beta D v w p hp.1.le)
theorem row_lintegral_tendsto
(hd : 3 ≤ d) {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hthreshold : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1)
(D : ℕ → Finset (TreeVertex d)) (hD : Monotone D)
(hconn : ∀ n, ConnectedFinset (D n)) (hex : ∀ u, ∃ n, u ∈ D n)
(v : TreeVertex d) (hv : ∀ n, v ∈ D n) (T : ℝ) :
Tendsto (fun n => ∫⁻ p, ∑' w, ENNReal.ofReal (rowError beta (D n) v w p ^ 2)
∂(volume.restrict (Ioc 0 T)).prod (observationLaw μ)) atTop (𝓝 0) := by
let := hμ.isProbabilityMeasure
simp_rw [rowError_lintegral_eq]
exact NormalizedInfiniteTimeEnergy.finiteRow_threshold_row_lintegral_tendsto
hd hμ hb hthreshold (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) D hD hconn hex v hv T
theorem rowError_integrable
{beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(D : Finset (TreeVertex d)) (hD : ConnectedFinset D)
(v : TreeVertex d) (hv : v ∈ D) {T : ℝ} (hT : 0 ≤ T) :
let P := (volume.restrict (Ioc 0 T)).prod (observationLaw μ)
(∀ᵐ p ∂P, Summable (fun w => rowError beta D v w p ^ 2)) ∧
Integrable (fun p => ∑' w, rowError beta D v w p ^ 2) P ∧
HasSum (fun w => ∫ p, rowError beta D v w p ^ 2 ∂P)
(∫ p, ∑' w, rowError beta D v w p ^ 2 ∂P) := by
let := hμ.isProbabilityMeasure
let P := (volume.restrict (Ioc 0 T)).prod (observationLaw μ)
have hf : (∫⁻ p, ∑' w, ENNReal.ofReal (rowError beta D v w p ^ 2) ∂P) ≠ ⊤ := by
rw [rowError_lintegral_eq]
exact NormalizedInfiniteTimeEnergy.lintegral_rowError_ne_top 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) D hD v hv hT
rw [lintegral_tsum (fun w =>
((measurable_rowError beta D v w).pow_const 2).ennreal_ofReal.aemeasurable)] at hf
have hm := fun w => (measurable_rowError beta D v w).aestronglyMeasurable (μ := P)
have hs := ae_summable_sq_of_tsum_lintegral_sq_ne_top P hm hf
have hi := integrable_tsum_sq_of_tsum_lintegral_sq_ne_top P hm hf
exact ⟨hs, hi, DiscrepancyEnergy.hasSum_integral_sq_row hm hs hi⟩
theorem row_integral_tendsto
(hd : 3 ≤ d) {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hthreshold : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1)
(D : ℕ → Finset (TreeVertex d)) (hD : Monotone D)
(hconn : ∀ n, ConnectedFinset (D n)) (hex : ∀ u, ∃ n, u ∈ D n)
(v : TreeVertex d) (hv : ∀ n, v ∈ D n) {T : ℝ} (hT : 0 ≤ T) :
let P := (volume.restrict (Ioc 0 T)).prod (observationLaw μ)
(∀ n, (∀ᵐ p ∂P, Summable (fun w => rowError beta (D n) v w p ^ 2)) ∧
Integrable (fun p => ∑' w, rowError beta (D n) v w p ^ 2) P ∧
HasSum (fun w => ∫ p, rowError beta (D n) v w p ^ 2 ∂P)
(∫ p, ∑' w, rowError beta (D n) v w p ^ 2 ∂P)) ∧
Tendsto (fun n => ∫ p, ∑' w, rowError beta (D n) v w p ^ 2 ∂P) atTop (𝓝 0) := by
let P := (volume.restrict (Ioc 0 T)).prod (observationLaw μ)
have hvalid (n : ℕ) := rowError_integrable hμ hb (D n) (hconn n) v (hv n) hT
refine ⟨hvalid, ?_⟩
have heq (n : ℕ) : (∫ p, ∑' w, rowError beta (D n) v w p ^ 2 ∂P) =
(∫⁻ p, ∑' w, ENNReal.ofReal (rowError beta (D n) v w p ^ 2) ∂P).toReal := by
rw [integral_eq_lintegral_of_nonneg_ae
(Eventually.of_forall fun p => tsum_nonneg fun w => sq_nonneg _)
(hvalid n).2.1.aestronglyMeasurable]
congr 1
apply lintegral_congr_ae
filter_upwards [(hvalid n).1] with p hp
exact ENNReal.ofReal_tsum_of_nonneg (fun w => sq_nonneg _) hp
have hlim := (ENNReal.tendsto_toReal (by simp : (0 : ℝ≥0∞) ≠ ⊤)).comp
(row_lintegral_tendsto hd hμ hb hthreshold D hD hconn hex v hv T)
change Tendsto (fun n => ∫ p, ∑' w, rowError beta (D n) v w p ^ 2 ∂P) atTop (𝓝 0)
simp_rw [heq]
exact hlim
theorem copies_row_lintegral_tendsto
(hd : 3 ≤ d) {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
[IsProbabilityMeasure μ]
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hthreshold : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1)
(D : ℕ → Finset (TreeVertex d)) (hD : Monotone D)
(hconn : ∀ n, ConnectedFinset (D n)) (hex : ∀ u, ∃ n, u ∈ D n)
(v : TreeVertex d) (hv : ∀ n, v ∈ D n) (j : Bool) (T : ℝ) :
Tendsto (fun n => ∫⁻ p, ∑' w, ENNReal.ofReal
(rowError beta (D n) v w (p.1, if j then p.2.2.1 else p.2.2.2) ^ 2)
∂(volume.restrict (Ioc 0 T)).prod (copies beta μ)) atTop (𝓝 0) := by
have : IsProbabilityMeasure ((observationLaw μ).map (innovationMap beta)) := inferInstance
let pr : Paths d × (Paths d × Paths d) → Paths d :=
fun q => if j then q.2.1 else q.2.2
have hpr : MeasurePreserving pr (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
have hp := (MeasurePreserving.id (volume.restrict (Ioc (0 : ℝ) T))).prod hpr
have heq (n : ℕ) : (∫⁻ p, ∑' w, ENNReal.ofReal
(rowError beta (D n) v w (p.1, if j then p.2.2.1 else p.2.2.2) ^ 2)
∂(volume.restrict (Ioc 0 T)).prod (copies beta μ)) =
(∫⁻ p, ∑' w, ENNReal.ofReal (rowError beta (D n) v w p ^ 2)
∂(volume.restrict (Ioc 0 T)).prod (observationLaw μ)) :=
hp.lintegral_comp (Measurable.tsum fun w =>
((measurable_rowError beta (D n) v w).pow_const 2).ennreal_ofReal)
simp_rw [heq]
exact row_lintegral_tendsto hd hμ hb hthreshold D hD hconn hex v hv T
theorem copies_row_integral_tendsto
(hd : 3 ≤ d) {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
[IsProbabilityMeasure μ]
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hthreshold : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1)
(D : ℕ → Finset (TreeVertex d)) (hD : Monotone D)
(hconn : ∀ n, ConnectedFinset (D n)) (hex : ∀ u, ∃ n, u ∈ D n)
(v : TreeVertex d) (hv : ∀ n, v ∈ D n) (j : Bool) {T : ℝ} (hT : 0 ≤ T) :
let P := (volume.restrict (Ioc 0 T)).prod (copies beta μ)
let E := fun n w p => rowError beta (D n) v w
(p.1, if j then p.2.2.1 else p.2.2.2)
(∀ n, (∀ᵐ p ∂P, Summable (fun w => E n w p ^ 2)) ∧
Integrable (fun p => ∑' w, E n w p ^ 2) P) ∧
Tendsto (fun n => ∫ p, ∑' w, E n w p ^ 2 ∂P) atTop (𝓝 0) := by
have : IsProbabilityMeasure ((observationLaw μ).map (innovationMap beta)) := inferInstance
let pr : Paths d × (Paths d × Paths d) → Paths d :=
fun q => if j then q.2.1 else q.2.2
have hpr : MeasurePreserving pr (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
have hp := (MeasurePreserving.id (volume.restrict (Ioc (0 : ℝ) T))).prod hpr
have ho := row_integral_tendsto hd hμ hb hthreshold D hD hconn hex v hv hT
refine ⟨fun n => ⟨hp.quasiMeasurePreserving.ae (ho.1 n).1,
hp.integrable_comp_of_integrable (ho.1 n).2.1⟩, ?_⟩
have heq (n : ℕ) : (∫ p, ∑' w,
rowError beta (D n) v w (p.1, if j then p.2.2.1 else p.2.2.2) ^ 2
∂(volume.restrict (Ioc 0 T)).prod (copies beta μ)) =
(∫ p, ∑' w, rowError beta (D n) v w p ^ 2
∂(volume.restrict (Ioc 0 T)).prod (observationLaw μ)) :=
hp.hasLaw.integral_comp (ho.1 n).2.1.aestronglyMeasurable
change Tendsto (fun n => ∫ p, ∑' w,
rowError beta (D n) v w (p.1, if j then p.2.2.1 else p.2.2.2) ^ 2
∂(volume.restrict (Ioc 0 T)).prod (copies beta μ)) atTop (𝓝 0)
simp_rw [heq]
exact ho.2
end Problem367.NormalizedObservationRows
end
end OAI
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