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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 | import OAI.Probability.FreeIsing.Cavity.Array
import OAI.Probability.FreeIsing.Estimates.CoupledSmallLogLoss
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
namespace Problem367.CavityArray
variable {d : ℕ} {theta : ℝ}
theorem field_identDistrib_of_measurePreserving
{Ω : Type*} [MeasurableSpace Ω] {ν : Measure Ω}
{μ : Measure (TreeVertex d → ℝ)} {F : Ω → (TreeVertex d → ℝ)}
(hF : MeasurePreserving F ν μ) (h0 : 0 ≤ theta) (h1 : theta < 1)
(v w : TreeVertex d) :
IdentDistrib (fun ω => field h0 h1 (F ω) v w)
(fun H => field h0 h1 H v w) ν μ := by
have hm := measurable_field_apply h0 h1 v w
refine ⟨(hm.comp hF.measurable).aemeasurable, hm.aemeasurable, ?_⟩
rw [show (fun ω => field h0 h1 (F ω) v w) =
(fun H => field h0 h1 H v w) ∘ F from rfl,
← Measure.map_map hm hF.measurable, hF.map_eq]
theorem integrable_site_sub_sq_of_bound
{ν : Measure (SitePair d)} [IsProbabilityMeasure ν]
{M : ℝ} (hM : 0 ≤ M) (v : TreeVertex d)
(hbound : ∀ᵐ p ∂ν, |p.1 v - p.2 v| ≤ M) :
Integrable (fun p : SitePair d => (p.1 v - p.2 v) ^ 2) ν := by
apply (integrable_const (M ^ 2)).mono'
((((measurable_pi_apply v).comp measurable_fst).sub
((measurable_pi_apply v).comp measurable_snd)).pow_const 2).aestronglyMeasurable
filter_upwards [hbound] with p hp
rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
simpa only [sq_abs, Function.comp_apply, Pi.sub_apply] using
(sq_le_sq₀ (abs_nonneg _) hM).2 hp
theorem integral_log_field_sub_sq_le
{ν : Measure (SitePair d)}
(h0 : 0 ≤ theta) (h1 : theta < 1)
(hinv : ∀ g : TreeAut d, MeasurePreserving (pairRelabel g) ν ν)
(o v w : TreeVertex d)
(hInt : Integrable (fun p : SitePair d => (p.1 o - p.2 o) ^ 2) ν) :
(∫ p, (Real.log (responseAttenuation theta (field h0 h1 p.1 v w)) -
Real.log (responseAttenuation theta (field h0 h1 p.2 v w))) ^ 2 ∂ν) ≤
cavityLogVarianceCoefficient theta *
(∫ p : SitePair d, (p.1 o - p.2 o) ^ 2 ∂ν) := by
have hv := (site_sub_identDistrib hinv o v).sq
have hw := (site_sub_identDistrib hinv o w).sq
exact integral_cavity_log_discrepancy_sq_le h0 h1
((measurable_field_apply h0 h1 v w).comp measurable_fst)
((measurable_field_apply h0 h1 v w).comp measurable_snd)
(hv.integrable_snd hInt) (hw.integrable_snd hInt)
(Filter.Eventually.of_forall (fun p =>
⟨field_spec h0 h1 p.1 v w, field_spec h0 h1 p.1 w v,
field_spec h0 h1 p.2 v w, field_spec h0 h1 p.2 w v⟩))
hv.integral_eq.symm.le hw.integral_eq.symm.le
theorem integral_log_field_sub_sq_small_time
{ν : Measure (SitePair d)} [IsProbabilityMeasure ν]
{μ : Measure (TreeVertex d → ℝ)}
(h0 : 0 ≤ theta) (h1 : theta < 1)
(hinv : ∀ g : TreeAut d, MeasurePreserving (pairRelabel g) ν ν)
(hfst : MeasurePreserving Prod.fst ν μ)
(hsnd : MeasurePreserving Prod.snd ν μ)
(o v w : TreeVertex d) {M t c : ℝ} (hM : 0 ≤ M) (ht : 0 < t)
(hbound : ∀ᵐ p ∂ν, ∀ z : TreeVertex d, |p.1 z - p.2 z| ≤ M)
(hTail : μ {H | t ^ (1 / 8 : ℝ) < |field h0 h1 H v w|} ≤
ENNReal.ofReal (Real.exp (-c * (Real.log (1 / t)) ^ 2))) :
(∫ p, (Real.log (responseAttenuation theta (field h0 h1 p.1 v w)) -
Real.log (responseAttenuation theta (field h0 h1 p.2 v w))) ^ 2 ∂ν) ≤
cavityLogVarianceCoefficient theta * t ^ (1 / 4 : ℝ) *
(∫ p : SitePair d, (p.1 o - p.2 o) ^ 2 ∂ν) +
min (cavityLogVarianceCoefficient theta *
(∫ p : SitePair d, (p.1 o - p.2 o) ^ 2 ∂ν))
(cavityLogTailCoefficient theta M * Real.exp (-c * (Real.log (1 / t)) ^ 2)) := by
have hInt := integrable_site_sub_sq_of_bound hM o (hbound.mono (fun _ h => h o))
have hv := (site_sub_identDistrib hinv o v).sq
have hw := (site_sub_identDistrib hinv o w).sq
exact integral_cavity_log_discrepancy_small_time_of_common_law h0 h1
((measurable_field_apply h0 h1 v w).comp measurable_fst)
((measurable_field_apply h0 h1 v w).comp measurable_snd)
(hv.integrable_snd hInt) (hw.integrable_snd hInt)
(Filter.Eventually.of_forall (fun p =>
⟨field_spec h0 h1 p.1 v w, field_spec h0 h1 p.1 w v,
field_spec h0 h1 p.2 v w, field_spec h0 h1 p.2 w v⟩))
hM ht (hbound.mono (fun _ h => ⟨h v, h w⟩))
hv.integral_eq.symm.le hw.integral_eq.symm.le
(field_identDistrib_of_measurePreserving hfst h0 h1 v w)
(field_identDistrib_of_measurePreserving hsnd h0 h1 v w) hTail
end Problem367.CavityArray
end
end OAI
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