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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 | import OAI.Probability.FreeIsing.Ising.CanonicalNormalizedField
import OAI.Probability.FreeIsing.Ising.CanonicalResponseIntegrals
import OAI.Probability.FreeIsing.Ising.CopiedDriftDecomposition
import OAI.Probability.FreeIsing.Ising.CopyResponseIdentification
import OAI.Probability.FreeIsing.Ising.CopySquareDensity
import OAI.Probability.FreeIsing.Ising.JoinedResponseIntegrals
import OAI.Probability.FreeIsing.Ising.StoppedSquareAssembly
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped NNReal Topology
namespace Problem367.IsingObservationFactor
open IsingObservationInnovation IsingCopyEnergy
variable {d : ℕ} {beta : ℝ}
variable {μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ]
theorem variance_analysis_of_joined_decomposition
(hd : 3 ≤ d) (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1)
(S : IsingJoinedResponseIntegrals.IntegralSystem hb μ)
(hrep : ∀ᵐ z ∂copies beta μ, ∀ t,
continuousDifference beta (ConnectedHull.root d) t z =
(S.total true t z - S.total false t z) +
nnrealTimePrimitive (copyResponseDriftDifference beta (ConnectedHull.root d)) t z)
(T : ℝ≥0) :
AbsolutelyContinuousOnInterval (fun t : ℝ => ∫ z,
(field beta (ConnectedHull.root d) t.toNNReal z.2.1 -
field beta (ConnectedHull.root d) t.toNNReal z.2.2) ^ 2 ∂copies beta μ) 0 T ∧
(∀ᵐ t ∂volume.restrict (Icc (0 : ℝ) T),
deriv (fun s : ℝ => ∫ z,
(field beta (ConnectedHull.root d) s.toNNReal z.2.1 -
field beta (ConnectedHull.root d) s.toNNReal z.2.2) ^ 2 ∂copies beta μ) t =
∫ z, density beta (ConnectedHull.root d) t z ∂copies beta μ) := by
have hM := (S.total_martingale true).sub (S.total_martingale false)
have h2 (t : ℝ≥0) := (S.total_memLp true t).sub (S.total_memLp false t)
have hc : ∀ᵐ z ∂copies beta μ, Continuous (fun t => S.total true t z - S.total false t z) := by
filter_upwards [S.total_continuous true, S.total_continuous false] with z hl hr
exact hl.sub hr
have hp := (S.total_predictable true).isStronglyProgressive.sub
(S.total_predictable false).isStronglyProgressive
have hQ : Martingale (fun t z => (S.total true t z - S.total false t z) ^ 2 -
nnrealTimePrimitive (copyResponseNoiseEnergy hb) t z)
ContinuousCopyBrownian.joinedFiltration (copies beta μ) :=
S.difference_compensatedSquare
apply actual_variance_analysis_of_common_local_square hd hμ hb hcrit (ConnectedHull.root d) T
intro n t ht
have hs := stopped_difference_energy_of_representation hd hμ hb hcrit hM h2 hc hp hQ hrep n
((isLocalizingSequence_commonPosteriorExit hd hμ hb hcrit (ConnectedHull.root d)).isStoppingTime n)
(fun _ => le_rfl) t
apply hs.trans
have hnoise : IsingCopySquareDensity.noise (d := d) hb = copyResponseNoiseEnergy hb := by
funext s z
rfl
have hdrift : IsingCopySquareDensity.drift (d := d) beta =
copyResponseDriftDifference beta (ConnectedHull.root d) := by
funext s z
rfl
simpa only [hnoise, hdrift, mul_assoc] using
IsingCopySquareDensity.integral_integral_stopped_noise_add_drift_eq_actual
hd hμ hb hcrit (commonPosteriorExit beta (ConnectedHull.root d) n) t
theorem actual_variance_analysis
(hd : 3 ≤ d) (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1)
(T : ℝ≥0) :
AbsolutelyContinuousOnInterval (fun t : ℝ => ∫ z,
(field beta (ConnectedHull.root d) t.toNNReal z.2.1 -
field beta (ConnectedHull.root d) t.toNNReal z.2.2) ^ 2 ∂copies beta μ) 0 T ∧
(∀ᵐ t ∂volume.restrict (Icc (0 : ℝ) T),
deriv (fun s : ℝ => ∫ z,
(field beta (ConnectedHull.root d) s.toNNReal z.2.1 -
field beta (ConnectedHull.root d) s.toNNReal z.2.2) ^ 2 ∂copies beta μ) t =
∫ z, density beta (ConnectedHull.root d) t z ∂copies beta μ) := by
obtain ⟨I, Y, hI2, hY2, hI, hIlim, _hYp, hYM, hYc, _hY0,
_hYiso, hYsum, _hYmax, _hQ, _hAint, _hAc⟩ :=
exists_canonical_root_response_integrals hd hμ hb hcrit
obtain ⟨S⟩ := IsingJoinedResponseIntegrals.exists_integrals hd hμ hb hcrit
have hfield := canonical_field_eq_normalized_integral hd hμ hb hcrit
(fun a => (hI a).2.1) hYM hI2 hY2 (fun a => (hI a).2.2) hYc hYsum hIlim
have hcopy := rootResponse_sum_eq_copy_all_times hμ hb
hI2 hY2 S.channel_memLp S.total_memLp hYc S.total_continuous
hIlim S.channel_limit hYsum S.total_sum
have hrep := copied_difference_decomposition hμ hb (ConnectedHull.root d)
hfield (fun j => hcopy.mono (fun z hz => hz j))
exact variance_analysis_of_joined_decomposition hd hμ hb hcrit S hrep T
end Problem367.IsingObservationFactor
end
end OAI
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