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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 | import OAI.Probability.FreeIsing.Ising.CanonicalPosteriorStops
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped NNReal
namespace Problem367.IsingObservationFactor
open IsingObservationInnovation IsingCopyEnergy PosteriorCompensatorStopping
variable {d : ℕ} {beta : ℝ}
variable {μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ]
def copyResponseDriftDifference (beta : ℝ) (v : TreeVertex d)
(t : ℝ≥0) (q : Joint d) : ℝ :=
posterior beta v t q.2.1 * copyResponseEnergy beta v true t q -
posterior beta v t q.2.2 * copyResponseEnergy beta v false t q
lemma copyResponseDriftDifference_progressive (beta : ℝ) (v : TreeVertex d) :
IsStronglyProgressive ContinuousCopyBrownian.joinedFiltration
(copyResponseDriftDifference beta v) := by
have hl := isStronglyProgressive_infiniteGibbsSpinMean
ContinuousCopyBrownian.joinedFiltration beta v _
IsingConditionalCopyBounds.progressive_leftObservation
have hr := isStronglyProgressive_infiniteGibbsSpinMean
ContinuousCopyBrownian.joinedFiltration beta v _
IsingConditionalCopyBounds.progressive_rightObservation
exact (hl.mul (copyResponseEnergy_progressive beta v true)).sub
(hr.mul (copyResponseEnergy_progressive beta v false))
lemma copyResponseEnergy_nonneg (beta : ℝ) (v : TreeVertex d) (j : Bool)
(t : ℝ≥0) (q : Joint d) : 0 ≤ copyResponseEnergy beta v j t q :=
tsum_nonneg (fun _ => sq_nonneg _)
theorem abs_copyResponseDriftDifference_le (beta : ℝ) (v : TreeVertex d)
(t : ℝ≥0) (q : Joint d) :
|copyResponseDriftDifference beta v t q| ≤
copyResponseEnergy beta v true t q + copyResponseEnergy beta v false t q := by
have hl : |posterior beta v t q.2.1| ≤ 1 := posterior_norm_le_one beta v t q.2.1
have hr : |posterior beta v t q.2.2| ≤ 1 := posterior_norm_le_one beta v t q.2.2
unfold copyResponseDriftDifference
calc
_ ≤ |posterior beta v t q.2.1 * copyResponseEnergy beta v true t q| +
|posterior beta v t q.2.2 * copyResponseEnergy beta v false t q| := abs_sub _ _
_ = |posterior beta v t q.2.1| * copyResponseEnergy beta v true t q +
|posterior beta v t q.2.2| * copyResponseEnergy beta v false t q := by
rw [abs_mul, abs_mul, abs_of_nonneg (copyResponseEnergy_nonneg beta v true t q),
abs_of_nonneg (copyResponseEnergy_nonneg beta v false t q)]
_ ≤ _ := add_le_add
(mul_le_of_le_one_left (copyResponseEnergy_nonneg beta v true t q) hl)
(mul_le_of_le_one_left (copyResponseEnergy_nonneg beta v false t q) hr)
theorem copyResponseDriftDifference_integrable
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(v : TreeVertex d) (T : ℝ≥0) :
Integrable (fun p : ℝ × Joint d => copyResponseDriftDifference beta v p.1.toNNReal p.2)
((volume.restrict (Ioc (0 : ℝ) T)).prod (copies beta μ)) := by
have hm := (stronglyMeasurable_joint_of_isStronglyProgressive
(copyResponseDriftDifference_progressive beta v)).comp_measurable
((measurable_real_toNNReal.comp measurable_fst).prodMk measurable_snd)
apply ((copyResponseEnergy_integrable hμ hb v true T).add
(copyResponseEnergy_integrable hμ hb v false T)).mono' hm.aestronglyMeasurable
exact Eventually.of_forall (fun p => abs_copyResponseDriftDifference_le beta v p.1.toNNReal p.2)
theorem ae_all_abs_driftPrimitive_le_commonResponseCompensator
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(v : TreeVertex d) :
∀ᵐ q ∂copies beta μ, ∀ t : ℝ≥0,
|nnrealTimePrimitive (copyResponseDriftDifference beta v) t q| ≤
commonResponseCompensator beta v t q := by
have hiB := ae_intervalIntegrable_horizons_of_joint
(μ := copies beta μ) (a := copyResponseDriftDifference beta v)
(fun n => by simpa only [NNReal.coe_natCast] using
copyResponseDriftDifference_integrable hμ hb v (n : ℝ≥0))
have hiE := ae_intervalIntegrable_horizons_of_joint
(μ := copies beta μ)
(a := fun t q => copyResponseEnergy beta v true t q + copyResponseEnergy beta v false t q)
(fun n => by
have hi := (copyResponseEnergy_integrable hμ hb v true (n : ℝ≥0)).add
(copyResponseEnergy_integrable hμ hb v false (n : ℝ≥0))
convert hi using 1; rfl)
filter_upwards [hiB, hiE] with q hB hE
intro t
obtain ⟨n, hn⟩ := exists_nat_ge (t : ℝ)
have hrestrict {f : ℝ → ℝ} (hi : IntervalIntegrable f volume 0 (n : ℝ)) :
IntervalIntegrable f volume 0 (t : ℝ) :=
(intervalIntegrable_iff_integrableOn_Ioc_of_le t.coe_nonneg).mpr
(((intervalIntegrable_iff_integrableOn_Ioc_of_le (Nat.cast_nonneg n)).mp hi).mono_set
(Ioc_subset_Ioc le_rfl hn))
unfold nnrealTimePrimitive commonResponseCompensator
calc
_ ≤ ∫ s in (0 : ℝ)..(t : ℝ), |copyResponseDriftDifference beta v s.toNNReal q| :=
intervalIntegral.abs_integral_le_integral_abs t.coe_nonneg
_ ≤ _ := intervalIntegral.integral_mono t.coe_nonneg (hrestrict (hB n)).abs
(hrestrict (hE n)) (fun s => abs_copyResponseDriftDifference_le beta v s.toNNReal q)
theorem ae_all_abs_commonResponseCompensator_stopped_le
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(v : TreeVertex d) (n : ℕ) {τ : Joint d → WithTop ℝ≥0}
(hτ : ∀ q, τ q ≤ commonPosteriorExit beta v n q) :
∀ᵐ q ∂copies beta μ, ∀ t : ℝ≥0,
|stoppedProcess (commonResponseCompensator beta v)
τ t q| ≤ (n : ℝ) + 1 := by
filter_upwards [commonResponseCompensator_continuous hμ hb v] with q hc
intro t
apply abs_stoppedProcess_le_of_le_exit hc
(by simp [commonResponseCompensator, nnrealTimePrimitive]; positivity) _ t
exact (hτ q).trans ((min_le_left _ _).trans (min_le_right _ _))
theorem ae_all_abs_stoppedDriftPrimitive_le
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(v : TreeVertex d) (n : ℕ) {τ : Joint d → WithTop ℝ≥0}
(hτ : ∀ q, τ q ≤ commonPosteriorExit beta v n q) :
∀ᵐ q ∂copies beta μ, ∀ t : ℝ≥0,
|stoppedProcess (nnrealTimePrimitive (copyResponseDriftDifference beta v))
τ t q| ≤ (n : ℝ) + 1 := by
filter_upwards [ae_all_abs_driftPrimitive_le_commonResponseCompensator hμ hb v,
ae_all_abs_commonResponseCompensator_stopped_le hμ hb v n hτ] with q hB hE
intro t
exact (hB _).trans ((le_abs_self _).trans (hE t))
theorem ae_all_abs_cutoffDriftPrimitive_le
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(v : TreeVertex d) (n : ℕ) {τ : Joint d → WithTop ℝ≥0}
(hτ : ∀ q, τ q ≤ commonPosteriorExit beta v n q) :
∀ᵐ q ∂copies beta μ, ∀ t : ℝ≥0,
|nnrealTimePrimitive
(StoppedTimePrimitive.stopDensity (copyResponseDriftDifference beta v)
τ) t q| ≤ (n : ℝ) + 1 := by
filter_upwards [ae_all_abs_stoppedDriftPrimitive_le hμ hb v n hτ] with q hq
intro t
rw [← StoppedTimePrimitive.stoppedProcess_primitive]
exact hq t
end Problem367.IsingObservationFactor
end
end OAI
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