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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.Estimates.ClippedArtanhWeight
import OAI.Probability.FreeIsing.Ising.CanonicalArtanh
import OAI.Probability.FreeIsing.Ising.ContinuousRootResponse
import OAI.Probability.FreeIsing.Stochastic.StoppedIntegralLocality
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal Topology
namespace Problem367.ClippedArtanhWeight
open ArtanhMartingaleFormula PosteriorCompensatorStopping PosteriorStoppingTimes
theorem weight_eq_stopped_before_exit {Ω : Type*}
{M A : ℝ≥0 → Ω → ℝ} {ω : Ω}
(hc : Continuous (fun t => M t ω)) (hzero : M 0 ω = 0)
(n : ℕ) (t : ℝ≥0) (ht : (t : WithTop ℝ≥0) ≤ jointExit M A n ω) :
weight (threshold n) (M t ω) =
firstWeight (stoppedProcess M (jointExit M A n) t ω) := by
have hb := abs_stopped_posterior_le (A := A) hc hzero n t
rw [stoppedProcess_eq_of_le ht] at hb ⊢
exact weight_eq_of_abs_le hb
end Problem367.ClippedArtanhWeight
namespace Problem367.IsingObservationFactor
open IsingObservationInnovation ArtanhMartingaleFormula
open PosteriorStoppingTimes PosteriorCompensatorStopping
def canonicalClippedWeight {d : ℕ} (beta : ℝ) (n : ℕ)
(t : ℝ≥0) (x : Paths d) : ℝ :=
ClippedArtanhWeight.weight (threshold n)
(continuousPosterior beta (ConnectedHull.root d) t x)
lemma canonicalClippedWeight_progressive {d : ℕ} (beta : ℝ) (n : ℕ) :
IsStronglyProgressive (filtration d) (canonicalClippedWeight beta n) := by
intro T
exact (ClippedArtanhWeight.continuous_weight (threshold_nonneg n) (threshold_lt_one n)).comp_stronglyMeasurable
(continuousPosterior_progressive beta (ConnectedHull.root d) T)
lemma canonicalClippedWeight_bound {d : ℕ} (beta : ℝ) (n : ℕ)
(t : ℝ≥0) (x : Paths d) :
|canonicalClippedWeight beta n t x| ≤ 1 / (1 - threshold n ^ 2) :=
ClippedArtanhWeight.abs_weight_le (threshold_nonneg n) (threshold_lt_one n) _
theorem canonicalClippedWeight_continuous_ae
{d : ℕ} (hd : 3 ≤ d) {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) (n : ℕ) :
∀ᵐ x ∂observationLaw μ, Continuous (fun t => canonicalClippedWeight beta n t x) := by
filter_upwards [(continuousPosterior_modification hd hμ hb hcrit (ConnectedHull.root d)).1]
with x hx
exact (ClippedArtanhWeight.continuous_weight (threshold_nonneg n) (threshold_lt_one n)).comp hx
theorem canonicalClippedWeight_mul_covariance_before_exit
{d : ℕ} (hd : 3 ≤ d) {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) (n : ℕ) :
∀ᵐ x ∂observationLaw μ, ∀ t : ℝ≥0,
(t : WithTop ℝ≥0) ≤ canonicalPosteriorExit beta (ConnectedHull.root d) n x →
∀ w, canonicalClippedWeight beta n t x * continuousRootCovariance hb w t x =
continuousRootResponse hb w t x := by
filter_upwards [(continuousPosterior_modification hd hμ hb hcrit (ConnectedHull.root d)).1,
continuousPosterior_initial_zero hd hμ hb hcrit (ConnectedHull.root d)] with x hc hz
intro t ht w
have hm := abs_stopped_posterior_le
(A := canonicalCovarianceCompensator beta (ConnectedHull.root d)) hc hz n t
change (t : WithTop ℝ≥0) ≤ jointExit
(continuousPosterior beta (ConnectedHull.root d))
(canonicalCovarianceCompensator beta (ConnectedHull.root d)) n x at ht
rw [stoppedProcess_eq_of_le ht] at hm
exact ClippedArtanhWeight.weight_mul_covariance_eq (threshold_nonneg n) (threshold_lt_one n) hm rfl
theorem canonicalClippedWeight_uniformSum_eq
{d : ℕ} (hd : 3 ≤ d) {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) (n : ℕ) (T : ℝ≥0) (k : ℕ) :
let M := continuousPosterior beta (ConnectedHull.root d)
let τ := canonicalPosteriorExit beta (ConnectedHull.root d) n
BrownianGrid.uniformSum (stoppedProcess M τ) (canonicalClippedWeight beta n) T k
=ᵐ[observationLaw μ]
BrownianGrid.uniformSum (stoppedProcess M τ)
(fun t x => firstWeight (stoppedProcess M τ t x)) T k := by
filter_upwards [(continuousPosterior_modification hd hμ hb hcrit (ConnectedHull.root d)).1,
continuousPosterior_initial_zero hd hμ hb hcrit (ConnectedHull.root d)] with x hc hz
apply StoppedIntegralLocality.uniformSum_stopped_congr
intro t ht
exact ClippedArtanhWeight.weight_eq_stopped_before_exit hc hz n t ht
end Problem367.IsingObservationFactor
end
end OAI
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